MATHEMATICS
FOR
MECHANICAL ENGINEERS

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MATHEMATICS
FOR
MECHANICAL ENGINEERS

Problems and Solutions

S. H. Omran
M. T. Chaichan
H. M. Hussen, PhD
N. G. Nacy, PhD
L. J. Habeeb, PhD

MERCURY LEARNING AND INFORMATION
Dulles, Virginia
Boston, Massachusetts
New Delhi

Copyright ©2021 by MERCURY LEARNING AND INFORMATION LLC. All rights reserved. Reprinted and revised with permission. Original title Mathematics for Mechanical Engineering: A Solutions Manual. Copyright ©2019 by S. K. Kataria and Sons. All rights reserved.

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S. H. Omran et al. Mathematics for Mechanical Engineers: Problems and Solutions.

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CONTENTS

Chapter 1:  The Rate of Change of a Function

Chapter 2:  Functions

Chapter 3:  Derivatives

Chapter 4:  Applications of Derivatives

Chapter 5:  Integration

Chapter 6:  Methods of Integration

Chapter 7:  Application of Integrals

Chapter 8:  Matrices and Determinants

Chapter 9:  Complex Numbers

CHAPTER 1

THE RATE OF CHANGE OF A FUNCTION

PROBLEMS

PROBLEM 1.1

The steel in railroad tracks expand when heated. For the track temperature encountered in normal outdoor use, the length S of a piece of track is related to its temperature t by a linear equation. An experiment with a piece of track gave the following measurements:

Write a linear equation for the relation between s and t.

Solution:

PROBLEM 1.2

Three of the following four points lie on a circle whose center is at the origin. What are they and what is the radius of the circle?

, and

Solution:

∴ A and B and D lie in the circle and radius .

PROBLEM 1.3

A and B are the points (3, 4) and (7, 1), respectively. Use Pythagorian theorem to prove that OA is perpendicular to AB. Calculate the slopes of OA and AB, and find their product.

Solution: The points are , and .

PROBLEM 1.4

, and S are the vertices of a parallelogram. Find the coordinates of M, and the point of intersection of the diagonals and of S.

Solution:

Coordinates of M are

Ans.

PROBLEM 1.5

Calculate the area of the triangle formed by the line
and the axes.

Solution:

The area of the triangle

PROBLEM 1.6

Find the equation of the straight line through perpendicular to the straight line AB whose equation is Calculate the length of the perpendicular from P and AB.

Solution:

Let Q be the point intersection of PQ and AB.

The slope of the line AB is …(1)

The slope of the line .

…(2)

Solving Equations (1) and (2), we obtain

By substituting this answer into Equation (2), we obtain

PROBLEM 1.7

, and are the mid-points of the sides BC, CA, and AB, respectively, of the triangle ABC. Find the equation of AB.

Solution: The coordinates of A, B, and C are

…(1)

…(2)

…(3)

…(4)

…(5)

…(6)

From (2),

From (3), …(7)

Subtracting, we obtain …(8)

∴    or    …(9)

By substituting this answer into Equation (1), we obtain

By substituting this answer into Equation (7), we obtain

PROBLEM 1.8

The straight line cuts the curve at P and Q. Calculate the length of PQ.

Solution:

PROBLEM 1.9

A line is drawn through the point (2, 3) making an angle of 45° with the positive direction of the x-axis, and it meets the line x = 6 at P. Find the distance of P from the origin O and the equation of the line through P perpendicular to OP.

Solution: Let us consider the point where the line through point (2, 3) and the line x = 6 meets at point (6, y).

The slope of the line passing through (2, 3) and (6, y) is .

The distance between the origin and the point is

To find the equation of the through P and perpendicular to OP, we solve the following:

PROBLEM 1.10

The vertices of a quadrilateral ABCD are , and . Prove that the diagonals AC and BD bisect each other at right angles, and that the length of BD is four times that of AC.

Solution: are the vertices of quadrilateral .

PROBLEM 1.11

The coordinates of vertices A, B and C of the triangle ABC are , and , respectively.

(a) Prove that the triangle is isosceles.

(b) Calculate the length of the perpendicular from B to AC, and use it to find the area of the triangle.

Solution:

(a)

∴ AB = AC ⇒ the triangle is isosceles

(b) Let D be the bisection between B and AC, D (2, 7).

PROBLEM 1.12

Find the equations of the lines that pass through the point of intersection of the lines and , and are respectively parallel and perpendicular to the line .

Solution:

By substituting this answer into Equation (2), we obtain

By substituting y into Equation (1), we obtain

∴ The point of intersection between two lines is .

The slope of line is .

The equation of a parallel line is and point .

∴ The equation of the required line is as follows:

The equation of perpendicular line is and point .

PROBLEM 1.13

Through the point A (1, 5), a line is drawn parallel to the x-axis to meet the line PQ at B, whose equation is . Find the length of AB and the sine of the angle between PQ and AB; hence, show that the length of the perpendicular from A to PQ is . Calculate the area of the triangle formed by PQ and the axes.

Solution: The line PQ is

…(1)

…(2)

(Since line AB ‖ to x-axis)

…(1)

…(2)

From Equation (1),

By substituting y into Equation (2), we obtain

By substituting y into Equation (2), we obtain

By substituting this answer into Equation (2), we obtain

To find the intersection PQ to the axis,

PROBLEM 1.14

Let express in terms of y and find the values of y for which x is real.

Solution:

PROBLEM 1.15

Find the domain and range of each function:

(a) (b) (c)

Solution: (a)

(b)

(c)

PROBLEM 1.16

Find the points of intersection of and .

Solution:

…(1)

…(2)

∴ The points of intersection are .

PROBLEM 1.17

Find the coordinates of the points at which the curves cut the axes:
(a) , (b) , (c)

Solution: (a)

∴ The curves cut the x-axis at .

(b)

At

∴ The curves cut the y-axis at .

∴ The curve cuts the x-axis at or .

(c)

At

∴ The curves cut the y-axis at .

∴ The curves cut the x-axis at or .

PROBLEM 1.18

Let and . What condition must be satisfied by the constants a, b, c, and d to make and indentical?

Solution:

PROBLEM 1.19

A particle moves in the plane from to the y-axis in such a way that . Find its new coordinates.

Solution:

Hence is the new coordinate in the (+) y-axis.

Hence is the new coordinate in the (−) y-axis.

PROBLEM 1.20

If and , what are the domains of f, g, f+g, f−g, f·g, f/g, g/f, fog, and gof ? What is the domain of

Solution:

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

PROBLEM 1.21

Discuss the continuity of the function.

Solution: At .

∴ The function is not continuous at .

∴ The function is continuous at .

∴ The function is not continuous at .

PROBLEM 1.22

Evaluate the following limits:

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Solution: (a)

or

(b)

(c)

(d)

(e)

(f)

(g)

PROBLEM 1.23

Suppose that and .
Find

(a) all zeros of f

(b) the value of k that makes h continuous at .

Solution:

(a)

(b)

CHAPTER 2

FUNCTIONS

PROBLEMS

PROBLEM 2.1

A body of unknown temperature was placed in a room that was held at . After 10 minutes, the body’s temperature was , and 20 minutes after the body was placed in the room, the body’s temperature was . Use Newton’s law of cooling to estimate the body’s initial temperature.

Solution:

when minute we have

…(1)

When , we have

… (2)

Divide Equation (2) by Equation (1):

… (3)

By substituting Equation (3) into Equation (1), we obtain

PROBLEM 2.2

A pan of warm water was put in a refrigerator. Ten minutes later, the water’s temperature was , and 10 minutes after that, it was . Use Newton’s law of cooling to estimate how cold the refrigerator was.

Ans.

Solution:

When, , we have

…(1)

When , we have

…(2)

By dividing Equation (2) by Equation (1), we obtain

By substituting this answer into Equation (1), we obtain

PROBLEM 2.3

Solve the following equations for values of from to inclusive:

(i)

(ii)

(iii)

(iv)

Solution: (i)

(ii)

or

(iii)

(iv)

PROBLEM 2.4

Solve the following equations for values of from to inclusive:

(i)

(ii)

(iii)

(iv)

Solution: (i)

(ii)

(iii)

(iv)

PROBLEM 2.5

If sin θ = 3/5, find without using tables the value of

(i)

(ii)

Solution:

PROBLEM 2.6

Find, without using tables, the values of and , when is .

Solution: As

PROBLEM 2.7

If and , where and are acute angles, find, without using tables, the values of

(a)

(b)

(c)

Solution:

(a) As we know that

(b)

(c)

PROBLEM 2.8

If and , where A is obtuse and B is acute, find, without using tables, the value of .

Solution:

PROBLEM 2.9

Prove the following identities:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

If A, B, C are angles of a triangle, show that

(viii)

(ix)

(x)

(xi)

(xii)

(xiii)

(xiv)

(xv)

(xvi)

and then verify

(xvii)

(xviii)

(xix)

(xx)

Solution: (i)

As

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

Since

(viii)

(ix)

(x)

(xi)

(xii)

(1)

Note

By substituting (a) and (b) in Equation (1),

(xiii)

[As ]

(xiv)

(xv)

(xvi) (a)

(b)

(xvii)

(xviii)

(xix)

(xx)

PROBLEM 2.10

If , prove that and deduce the formula for in terms of .

Solution:

prove that

PROBLEM 2.11

If , prove that .

Solution:

Dividing throughout by , we obtain

PROBLEM 2.12

If prove that .

Solution:

PROBLEM 2.13

If and show that

(i)

(ii)

Solution: (i)

(ii)

PROBLEM 2.14

If , prove that

Solution:

PROBLEM 2.15

If and , simplify:

(i)

(ii)

(iii)

Solution: (i)

(ii)

(iii)

PROBLEM 2.16

Eliminate from the following equations:

(i)

and

(ii)

and

(iii)

and

(iv)

and

Solution: (i) and

(ii)

After adding, we obtain After subtracting, we obtain

(iii)

On adding, we obtain

On subtracting, we obtain

Since,

(iv) and

PROBLEM 2.17

In the acute-angled triangle , the altitude makes angles and with with . Show by means of areas that if .

Solution:

PROBLEM 2.18

Given that , find , and .

Solution: Since,

PROBLEM 2.19

Evaluate the following expressions:

(a)

(b)

(c)

(d)

(e)

(f)

Solution:

(a)

(b)

(c)

(d)

(e)

(f)

PROBLEM 2.20

Find the angle in the graph (Hint: ).

Solution:

PROBLEM 2.21

Let sech ; determine the values of the remaining five hyperbolic functions.

Solution:

PROBLEM 2.22

Rewrite the following expressions in terms of exponentials; write the final result as simply as you can:

(a)

(b)

(c)

(d)

Solution:

(a)

(b)

(c)

(d)

PROBLEM 2.23

Solve the equation for x; .

Solution:

PROBLEM 2.24

Show that the distance from the origin to the point on the hyperbola is .

Solution:

PROBLEM 2.25

lies in the interval and . Show that , and .

Solution: (a) since

(b)

(c)

(d)

(e)

PROBLEM 2.26

Derive the formula:

Solution:

PROBLEM 2.27

Find :

Solution:

CHAPTER 3

DERIVATIVES

PROBLEMS

PROBLEM 3.1

Find for the following functions:

Solution:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

45.

PROBLEM 3.2

Verify the following derivatives:

(a)

(b)

Solution:

(a)

(b)

PROBLEM 3.3

Find the derivative of with respect to in the following functions:

(a)

(b)

Solution:

(a)

Now,

[Ans. ]

(b) and

[Ans. ]

PROBLEM 3.4

Find the second derivative for the following functions:

(a)

(b) at

(c)

Solution:

(a)

[Ans. ]

(b) at

[Ans. ]

(c)

[Ans. 6]

PROBLEM 3.5

Find the third derivative of the function:

Solution:

PROBLEM 3.6

Show for that

Solution:

PROBLEM 3.7

Show for that .

Solution:

PROBLEM 3.8

Show that satisfies .

Solution:

PROBLEM 3.9

Find for the following implicit functions:

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Solution:

(a)

On differentiating with respect to x,

(b)

(c)

On differentiating with respect to x, we obtain

(d)

(e)

(f)

On differentiating, we obtain

(g)

PROBLEM 3.10

Prove the following formulas:

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Solution:

(a)

(b)

(c)

Let,

(d)

(e)

(f)

(g)

(h)

PROBLEM 3.11

Show that the tangent to the hyperbola at the point , cuts the -axis at the point and except when vertical, cuts the -axis at the point .

Solution:

The slope at is

CHAPTER 4

APPLICATIONS OF DERIVATIVES

PROBLEMS

PROBLEM 4.1

Find the velocity if a particle’s position at time is .
When does the velocity vanish? Solution:

(Ans. 5.625)

PROBLEM 4.2

If a ball is thrown straight up with a velocity of its height after is given by the equation . At what instant will the ball be at its highest point? And how high will it rise?

Solution:

(Ans. 1, 16)

PROBLEM 4.3

A stone is thrown vertically upwards at . Its height is in meters above the point of projection, where is time in seconds later.

(a) What is the distance moved, and the average velocity during the (from to )?

(b) Find the average velocity for the intervals to , to ; and to .

(c) Deduce the actual velocity at the end of the .

Solution:

(a)

(b)

(c) At the end, the height

(Ans. (a) ; (b) , (c)

PROBLEM 4.4

A stone is thrown vertically upwards at from a point just a little higher than cliff’s ledge. Its height above the ledge later is . If its velocity is , differentiate to find in terms of :

(i)

When is the stone at the ledge level?

(ii)

Find its height and velocity after and 6 seconds.

(iii)

What meaning is attached to a negative value of ? A negative value of ?

(iv)

When is the stone momentarily at rest? What is the greatest height reached?

(v)

Find the total distance moved during the second.

Solution:

Average rate of change

(i) when

(ii)

(iii) The negative valve of S means that the stone is below ledge; the negative valve for v means a storm is blowing.

(iv)

(v) , the total distance traveled during the third second.

(Ans. ; (i) ; (ii) ;
(iii) ; (iv) ; (v) )

PROBLEM 4.5

A stone is thrown vertically downwards with a velocity of , and gravity produces on it an acceleration of :
(a) What is the velocity after ?
(b) Sketch the velocity-time graph.

Solution:

The velocity after time .

(Ans. )

PROBLEM 4.6

A car accelerates from to in 10 sec. Express this acceleration in (i) (ii) and (iii).

Solution:

(i)

(ii)

(iii)

(Ans. (i) ; (ii) ; (iii) )

PROBLEM 4.7

A car can accelerate at . How long will it take to reach from rest?

Solution:

(Ans. )

PROBLEM 4.8

An express train reducing its velocity to has to apply the brakes for 50 sec. If the retardation produced is , find its initial velocity in .

Solution:

(Ans. )

PROBLEM 4.9

At the instant from which time is measured, a particle is passing through and traveling towards , along the straight line . It is , from after sec, where :

(i)

When is it again at ?

(ii)

When and where is it momentarily at rest?

(iii)

What is the particle’s greatest displacement from , and how far does it move during the first ?

(iv)

What is the average velocity during the second?

(v)

At the end of the second, where is the particle, which way is it going, and is its speed increasing or decreasing?

(vi)

Repeat for the instant when .

Solution:

(i)

At , for the particle at ,

(ii)

Or

(iii)The particle’s greatest displacement from is during the first .

(iv)

(v)

(vi)At from (i.e., from on ); it is going from to the negative side of .

Since and , the speed is decreasing.

(Ans. (i) ; (ii) ; (iii) ; (iv) ;
(v) ; inceasing; (vi) ; decreasing)

PROBLEM 4.10

A particle moves in a straight line so that after sec it is , from a fixed point on the line, where . Find
(i) the acceleration when , and .
(ii) the average acceleration between and .

Solution:

(i)

(ii)

(Ans. (i) ; (ii) )

PROBLEM 4.11

A particle moves along the -axis in such a way that its distance from the origin after sec is given by the formula , what are its velocity and acceleration after sec? How long does it take for the velocity to be reduced from to , and how far does the particle travel mean while?

Solution:

(Ans. ; )

PROBLEM 4.12

A point moves along a straight line so that its distance from the point at time is given by the formula . Find:

(i)

at what times and in what positions the point will have zero velocity.

(ii)

its acceleration at these instants.

(iii)

its velocity when its acceleration is zero.

Solution:

(i)

(ii)

(iii)

(Ans. (i) (ii) (iii) )

PROBLEM 4.13

A particle moves in a straight line so that its distance from a fixed point on the line is given by , where is the time in seconds measured from . Find the speed of the particle when . Also find the distance from of the particle when , and show that it is then moving towards .

Solution:

, hence the particle is moving towards .

(Ans. )

PROBLEM 4.14

Find the limits for the following functions using L’Hopital’s rule:

(1)    (2)

(3)     (4)

(5)    (6)

(7)   (8)

(9)   (10)

Solution:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10. use L‘Hopital’s rule

PROBLEM 4.15

Find any local maximum and local minimum values, then sketch each curve by using the first derivative.

(1) (Ans. )

(2) (Ans. min. )

(3) (Ans. max. ; min. )

(4) (Ans. min. )

Solution:

(1)

The function has a local max at and local min at .

(2)

(3)

The function has a local max at and local min at .

(4)

The function has local minimum at .

PROBLEM 4.16

Find the interval of the -values on which the curve is concave up and concave down, then sketch the curve:
(1)
(2)
(3)
(4)

Solution:

(1)

(Ans. up down

The interval of the -values in which the curve is concave up at and concave down at .

(2)

is concave up when the value is up

(Ans. up )

(3)

The interval of the -values in which the curve is

Concave up at and concave down at .

(4)

The intervals of -values which the curve is

concave up and concave down

PROBLEM 4.17

Sketch the following curve by using the second derivative:

(1) (2) (3) (4)

Solution:

(1)

(Ans. max. ; min.

(2)

(Ans.

(3)

(Ans. max. )

(4)

(Ans. )

PROBLEM 4.18

What is the smallest perimeter possible for a rectangle of area ?

Solution:

is the length of the rectangle.

is the width of the rectangle.

The smallest perimeter is .

(Ans. 16)

PROBLEM 4.19

Find the area of the largest rectangle with its lower base on the
-axis and upper vertices on the parabola

Solution: The length of the rectangle is

The width is , and the area of .

The parabola is .

The area of the largest rectangle is .

(Ans. 32)

PROBLEM 4.20

A rectangular plot is to be bounded on one side by a straight river and enclosed on the other three sides by a fence. You have of fence at your disposal. What is the largest area you can enclose?

Solution:

is the width of the plot.

is the length of the plot.

The length of the fence is

The largest area is .

(Ans. 80000)

PROBLEM 4.21

Show that the rectangle that has the maximum area for a given perimeter is a square.

Solution:

is the length and is the width of the rectangle.

The perimeter is .

The area is .

The maximum area of the rectangle of a given perimeter is a square

where   .

PROBLEM 4.22

A wire of length is available for making a circle and a square. How should the wire be divided between the two shapes to maximize the sum of the enclosed areas?

Solution: is the radius of the circle.
is the length of the square.

The wire length is .

The total area is .

Hence, the maximum value of on the endpoints of the internal area is

Since

Hence, the wire should not be cut at all, but should be bent into a circle.

(Ans. All the wire should be bent into a circle.)

PROBLEM 4.23

A closed container is made from a right circular cylinder of radius and height with a hemispherical dome on top. Find the relationship between and that maximizes the volume for a given surface area .

Solution:

(Ans. )

PROBLEM 4.24

An open rectangular box is to be made from a piece of cardboard 8 in wide and 15 in long by cutting a square from each corner and bending up the sides. Find the dimensions of the box of largest volume.

Solution: is the length and is the width and the height of the rectangular box.

The length of the cutting square is .

(Ans. ; ; )

CHAPTER 5

INTEGRATION

PROBLEMS

PROBLEM 5.1

Evaluate

Solution:

(Ans. )

PROBLEM 5.2

Evaluate

Solution:

(Ans. )

PROBLEM 5.3

Evaluate

Solution:

(Ans. )

PROBLEM 5.4

Evaluate

Solution:

(Ans. )

PROBLEM 5.5

Evaluate

Solution:

(Ans. )

PROBLEM 5.6

Evaluate

Solution:

(Ans. )

PROBLEM 5.7

Evaluate

Solution:

(Ans. )

PROBLEM 5.8

Evaluate

Solution:

(Ans. )

PROBLEM 5.9

Evaluate

Solution: Here,

(Ans. )

PROBLEM 5.10

Evaluate

Solution:

Here,

(Ans. )

PROBLEM 5.11

Evaluate

Solution:

As,

(Ans. )

PROBLEM 5.12

Evaluate

Solution:

(Ans. )

PROBLEM 5.13

Evaluate

Solution:

(Ans. )

PROBLEM 5.14

Evaluate , where and are constants.

Solution:

(Ans. )

PROBLEM 5.15

Evaluate

Solution:

(Ans. )

PROBLEM 5.16

Evaluate

Solution:

(Ans. )

PROBLEM 5.17

Evaluate

Solution:

(Ans. )

PROBLEM 5.18

Evaluate

Solution:

(Ans. )

PROBLEM 5.19

Evaluate

Solution:

(Ans. )

PROBLEM 5.20

Evaluate

Solution:

(Ans. )

PROBLEM 5.21

Evaluate

Solution:

PROBLEM 5.22

Evaluate

Solution:

(Ans. )

PROBLEM 5.23

Evaluate

Solution:

(Ans. )

PROBLEM 5.24

Evaluate

Solution:

(Ans. )

PROBLEM 5.25

Evaluate

Solution:

(Ans. )

PROBLEM 5.26

Evaluate

Solution:

(Ans. )

PROBLEM 5.27

Evaluate

Solution:

(Ans. )

PROBLEM 5.28

Evaluate

Solution:

(Ans. )

PROBLEM 5.29

Evaluate

Solution:

(Ans. )

PROBLEM 5.30

Evaluate

Solution:

(Ans. )

PROBLEM 5.31

Evaluate

Solution:

(Ans. )

PROBLEM 5.32

Evaluate

Solution:

(Ans. )

PROBLEM 5.33

Evaluate

Solution:

(Ans. )

PROBLEM 5.34

Evaluate

Solution:

(Ans. )

PROBLEM 5.35

Evaluate

Solution:

(Ans. )

PROBLEM 5.36

Evaluate

Solution:

(Ans. )

PROBLEM 5.37

Evaluate

Solution:

(Ans. )

PROBLEM 5.38

Evaluate

Solution:

(Ans. )

PROBLEM 5.39

Evalute

Solution:

(Ans. )

PROBLEM 5.40

Evaluate

Solution:

(Ans. )

PROBLEM 5.41

Evaluate

Solution:

(Ans. )

PROBLEM 5.42

Evaluate

Solution:

(Ans. )

PROBLEM 5.43

Evaluate

Solution:

Since,

(Ans. )

PROBLEM 5.44

Evaluate

Solution:

(Ans. )

PROBLEM 5.45

Evaluate

Solution:

(Ans. )

PROBLEM 5.46

Evaluate

Solution:

(Ans. )

PROBLEM 5.47

Evaluate

Solution:

(Ans. )

PROBLEM 5.48

Evaluate

Solution:

(Ans. )

PROBLEM 5.49

Evaluate

Solution:

(Ans. )

PROBLEM 5.50

Evaluate

Solution:

(Ans. )

PROBLEM 5.51

Evaluate

Solution:

(Ans. )

PROBLEM 5.52

Evaluate

Solution:

(Ans. )

PROBLEM 5.53

Evaluate

Solution:

(Ans. )

PROBLEM 5.54

Evaluate

Solution:

(Ans. )

PROBLEM 5.55

Evaluate

Solution:

Since

(Ans. )

PROBLEM 5.56

Evaluate

Solution:

As

(Ans. )

PROBLEM 5.57

Evaluate

Solution:

(Ans. )

PROBLEM 5.58

Evaluate

Solution:

(Ans. )

PROBLEM 5.59

Evaluate

Solution:

(Ans. )

PROBLEM 5.60

Evaluate

Solution:

(Ans. )

PROBLEM 5.61

Evaluate

Solution:

(Ans. )

PROBLEM 5.62

Evaluate

Solution:

(Ans. )

PROBLEM 5.63

Evaluate

Solution:

(Ans. )

PROBLEM 5.64

Evaluate

Solution:

(Ans. )

PROBLEM 5.65

Evaluate

Solution:

(Ans. )

PROBLEM 5.66

Evaluate

Solution:

(Ans. )

PROBLEM 5.67

Evaluate

Solution:

If

Then,

(Ans. )

CHAPTER 6

METHODS OF INTEGRATION

PROBLEMS

PROBLEM 6.1

Evaluate

Solution:

(Ans. )

PROBLEM 6.2

Evaluate

Solution:

(Ans. )

PROBLEM 6.3

Evaluate

Solution:

(Ans. )

PROBLEM 6.4

Evaluate

Solution:

(Ans. )

PROBLEM 6.5

Evaluate

Solution:

Integrating by parts, we obtain

PROBLEM 6.6

Evaluate

Solution:

Integrate by partial fraction

(Ans. )

PROBLEM 6.7

Evaluate

Solution:

(Ans. )

PROBLEM 6.8

Evaluate

Solution:

Integration by partial fraction

From Equation (2) and Equation (4),

From Equation (1) and Equation (3),

(Ans. )

PROBLEM 6.9

Evaluate

Solution:

Use by part integration:

Let

(Ans. )

PROBLEM 6.10

Evaluate

Solution:

(Ans. )

PROBLEM 6.11

Evaluate

Solution:

Integration by parts

(Ans. )

PROBLEM 6.12

Evaluate

Solution:

Integration by parts

(Ans. )

PROBLEM 6.13

Evaluate

Solution:

Use integration by parts:

Integration by parts

(Ans. )

PROBLEM 6.14

Evaluate

Solution:

Integration by parts

Integration by parts

(Ans. )

PROBLEM 6.15

Evaluate

Solution:

Integration by parts

(Ans. )

PROBLEM 6.16

Evaluate

Solution:

Use integration by parts:

Integration by parts

Let

(Ans. )

PROBLEM 6.17

Evaluate

Solution:

(Ans. )

PROBLEM 6.18

Evaluate

Solution:

(Ans. )

PROBLEM 6.19

Evaluate

Solution:

Integration by parts

(Ans. )

PROBLEM 6.20

Evaluate

Solution:

Use integration by parts:

(Ans. )

PROBLEM 6.21

Evaluate

Solution:

(Ans. )

PROBLEM 6.22

Evaluate

Solution:

… (A)

Integration by parts

(Ans. )

PROBLEM 6.23

Evaluate

Solution:

(Ans. )

PROBLEM 6.24

Evaluate

Solution:

(Ans. )

PROBLEM 6.25

Evaluate

Solution:

(Ans. )

PROBLEM 6.26

Evaluate

Solution:

(Ans. )

PROBLEM 6.27

Evaluate

Solution:

Use integration by parts

(Ans. )

PROBLEM 6.28

Evaluate

Solution:

(Ans. )

PROBLEM 6.29

Evaluate

Solution:

(Ans. )

PROBLEM 6.30

Evaluate

Solution:

(Ans.

PROBLEM 6.31

Evaluate

Solution:

As

(Ans. )

PROBLEM 6.32

Evaluate

Solution:

Integration by parts

Let

Integration by parts

Let

(Ans. )

PROBLEM 6.33

Evaluate

Solution:

Integration by partial fraction

(Ans. )

PROBLEM 6.34

Evaluate

Solution:

(Ans. )

PROBLEM 6.35

Evaluate

Solution:

Integration by partial fraction

(Ans. )

PROBLEM 6.36

Evaluate

Solution:

(Ans. )

PROBLEM 6.37

Evaluate

Solution:

(Ans. )

PROBLEM 6.38

Evaluate

Solution:

(Ans. )

PROBLEM 6.39

Evaluate

Solution:

Let

Use integration by parts:

To find .

(Ans. )

PROBLEM 6.40

Evaluate

Solution:

(Ans. )

PROBLEM 6.41

Evaluate

Solution:

(Ans. )

PROBLEM 6.42

Evaluate

Solution:

PROBLEM 6.43

Evaluate

Solution:

As,

Let

(Ans. )

PROBLEM 6.44

Evaluate

Solution:

Let

We write

(Ans. )

PROBLEM 6.45

Evaluate

Solution:

Let

Integration by partial fraction

Substitute Equation (1) into Equation (3):

(Ans. )

PROBLEM 6.46

Evaluate

Solution:

As

Let

(Ans. )

PROBLEM 6.47

Evaluate

Solution:

Use integration by partial fraction.

Let

To find

,

let

(Ans.

PROBLEM 6.48

Evaluate

Solution:

Use integration by partial fraction

Let

(Ans. )

PROBLEM 6.49

Evaluate

Solution:

Let

(Ans. )

PROBLEM 6.50

Evaluate

Solution:

where

(Ans. )

PROBLEM 6.51

Evaluate

Solution:

Use integration by partial fraction.

(Ans. )

PROBLEM 6.52

Evaluate

Solution:

Integration by part

Let

To find

(Ans. )

PROBLEM 6.53

Evaluate

Solution:

Integration by parts

To find

(Ans. )

PROBLEM 6.54

Evaluate

Solution:

(Ans. )

CHAPTER 7

APPLICATION OF INTEGRALS

PROBLEMS

PROBLEM 7.1

Find the area of the region bounded by the given curves and lines for the following problems:

1. The coordinate axes and the line

2. The -axis and the curve and the lines

3. The curve and the line

4. The curves and

5. The parabola and the line .

Solution:

1. The coordinate axes and the line

2. The -axis and the curve and the lines

0

1

1

2.78

1.5

4.48

3. The curve and the line

4. The curves and

0

(0, 0)

0.5

0.7

0

0

1

1

0.75

0.5

0.91

0.75

0.96

1.2

5. The parabola and the line

(Ans. )

PROBLEM 7.2

Write an equivalent double integral with order of integration reversed for each integrals check your answer by evaluation both double integrals, and sketch the region.

1.   2.

3.

Solution:

1.

x

0

1

1

27.8

1.5

4.48

To find use integration by part

1

0

2

0.69

3

1.09

4

1.38

.

.

.

.

739

2

2.

0

0

0.5

0.707

0.75

0.866

1

1

0

0

1

1

2

4

3

9

(Ans. )

3.

2

0

1.4

1

0

0

1

0

0

1

1.22

2

0

0

1.22

(Ans. )

PROBLEM 7.3

Find the volume of the tetrahedron bounded by the plane and the coordinate planes.

Solution:

PROBLEM 7.4

Find the volume bounded by the plane laterally by the elliptic cylinder and above by the plane

Solution:

(Ans. )

PROBLEM 7.5

Find the lengthed of the following curves:

1.

2.

3.

4.

Solution:

1.

The position of the curve between the origin and point

(Ans. )

2.

(Ans. )

3. from to

(Ans. )

4. from to .

(Ans. )

PROBLEM 7.6

Find the distance travelled by the particle between and if the position at time is given by :

Solution:

(Ans. 12)

PROBLEM 7.7

The position of a particle at time is given by: . Find the distance it travel between and .

Solution:

(Ans. )

PROBLEM 7.8

Find the area of the surface generated by rotating about the -axis the arc of the curve between and

Solution:

(Ans. )

PROBLEM 7.9

Find the area of the surface generated by rotating about the -axis the arc of the curve between and .

Solution:

(Ans. )

PROBLEM 7.10

Find the area of the surface generated by rotating about the -axis the curve

Solution:

(Ans. )

PROBLEM 7.11

The curve described by the particle from to is rotated about the -axis. Find the surface area that is generated.

Solution:

(Ans. )

CHAPTER 8

MATRICES AND DETERMINANTS

PROBLEM

PROBLEM 8.1

Let Find

(a)  (b)  (c)  (d)  (e)

(f )  (g)  (h)  (i)

Solution:

Given

(a)

(Ans. )

(b)

(Ans. )

(c)

(Ans. )

(d)

(Ans. )

(e)

(Ans. )

(f)

(Ans. )

(g) (Ans. )

(h) (Ans. )

(i)

(Ans. )

PROBLEM 8.2

Find the value of if

Solution:

Find the value of .

(Ans. )

PROBLEM 8.3

Find and if

Solution:

PROBLEM 8.4

Let , . Find

(a)   (b)

Solution:

PROBLEM 8.5

Let . Find and show that .

Solution:

(a)

PROBLEM 8.6

For what value of will ?

Solution: Expand the determinant

(Ans. )

PROBLEM 8.7

Let be an arbitrary 3 by 3 matrix, and let be the matrix obtained from the 3 by 3 identity matrix by interchanging rows 1 and 2: . (a) Compute and show that you would get the same result by interchanging rows 1 and 2 of . (b) Compute and show that the result is what you obtain by interchanging columns 1 and 2 of .

Solution:

(a)

(b)

PROBLEM 8.8

Solve the following determinants:

(a)  (b)  (c)

(d)  (e)  (f)

(g)  (h)

Solution:

(a)

(b)

(c)

(d)

(e)

(f)

Expend this along the first column:

(g)

(h)

PROBLEM 8.9

Solve the following systems of equations:

(a)  (b)  (c)

(d)  (e)  (f)

(g)  (h)

Solution:

(a)

(Ans. )

(Ans. )

(c) Using Crammer’s rule,

(Ans. )

(d)

(Ans. )

(e)

(Ans. )

(f)

(Ans. )

(g)

The augmented matrix is

(h)

(Ans. )

CHAPTER 9

COMPLEX NUMBERS

PROBLEMS

PROBLEM 9.1

Find the values of

(a)   (b)   (c)

Solution: (a)

(Ans. )

(b)

(Ans. )

(c)

(Ans. )

PROBLEM 9.2

Show that for all combinations of signs.

Solution:

PROBLEM 9.3

Solve the following equation for the real numbers and :

Solution:

From Equation (1)

(Ans. )

PROBLEM 9.4

Show that .

Solution:

PROBLEM 9.5

Let and denote, respectively, the real and imaginary parts of ,
and show that

(a)

(b)

(c)

Solution:

(a)

(b)

(c)

(Ans. On the line )

PROBLEM 9.6

Graph the points that satisfy the given conditions:

(a)   (b)    (c)

Solution:

(a)

The center of circle

(Ans. On the circle with center , radius 2)

(b)

(Ans. On the circle with center , radius 1)

(c)

0

0

1

2

3

1

2

3

(Ans. On the line )

PROBLEM 9.7

Express the following complex number in exponential form with and .

(a)  (b)  (c)   (d)

Solution:

(a)

(b)

(c)

(d)

PROBLEM 9.8

Find the three cube roots of 1.

Solution:

PROBLEM 9.9

Find the two square roots of .

Solution:

PROBLEM 9.10

Find the three cube roots of .

Solution:

PROBLEM 9.11

Find the six sixth roots of (64).

Solution:

PROBLEM 9.12

Find the six Solutions of the equation: .

Solution:

PROBLEM 9.13

Find all Solutions of the equation .

Solution:

PROBLEM 9.14

Solve the equation .

Solution: