# MCQ in Integral Calculus Part 1 | Mathematics Board Exam

(Last Updated On: January 12, 2021) This is the Multiple Choice Questions Part 1 of the Series in Integral Calculus topic in Engineering Mathematics. In Preparation for the ECE Board Exam make sure to expose yourself and familiarize in each and every questions compiled here taken from various sources including but not limited to past Board Examination Questions in Engineering Mathematics, Mathematics Books, Journals and other Mathematics References.

#### MCQ Topic Outline included in Mathematics Board Exam Syllabi

• MCQ in Basic Integrals | MCQ in Integrals of Exponential Functions | MCQ in Integrals of Logarithmic Functions | MCQs in Integrals of Trigonometric Functions | MCQ in Integrals in Inverse Trigonometric Functions | MCQ in Integrals of Hyperbolic Functions | MCQs in Integrals of Trigonometric Substitution | MCQs in Integration by parts | MCQs in Integral involving Plane Areas | MCQ in Integral involving Centroid | MCQ in Integral involving Length of Arc | MCQs in Integral involving Propositions of Pappus | MCQ in Integral involving Work | MCQ in Integral involving Moment of Inertia

#### Start Practice Exam Test Questions Part 1 of the Series

Choose the letter of the best answer in each questions.

Problem 1: ECE Board April 1999

What is the integral of (3t – 1)3 dt?

A. (1/12)(3t – 1)4 + C

B. (1/12)(3t – 4)4 + C

C. (1/4)(3t – 1)4 + C

D. (1/4)(3t – 1)3 + C

Solution: Review Solution for Number 1

Problem 2: ECE Board November 1998

Evaluate the integral of dx / (x + 2) from -6 to -10.

A. 21/2

B. 1/2

C. ln 3

D. ln 2

Solution: Review Solution for Number 2

Problem 3: ECE Board November 1998, ME Board April 1998

Integrate xcos (2x2 + 7) dx.

A. (1/4)sin (2x2 + 7) + C

B. (1/4)cos (2x2 + 7) + C

C. ((sin θ) / 4(x2 + 7)) + C

D. sin (2x2 + 7) + C

Solution: Review Solution for Number 3

Problem 4: ME Board April 1995, ME Board April 1997

Integrate: (7x3 + 4x2) dx

A. (7x3/3) + (4x2/2) + C

B. (7x4/4) + (4x2/5) + C

C. (7x4/4) + (4x3/3) + C

D. 7x4 + (4x2/2) + C

Solution: Review Solution for Number 4

Problem 5: CE Board November 1995

What is the integral of sin5 x cos3 x dx if the lower limit is zero and the upper limit is π/2?

A. 0.0203

B. 0.0307

C. 0.0417

D. 0.0543

Solution: Review Solution for Number 5

Problem 6: CE Board November 1994

What is the integral of sin5 x dx if the lower limit is 0 and the upper limit is π/2?

A. 0.233

B. 0.333

C. 0.433

D. 0.533

Solution: Review Solution for Number 6

Problem 7: CE Board May 1996

Find the integral of 12 sin5 x cos5 x dx if the lower limit is 0 and the upper limit is π/2.

A. 0.2

B. 0.333

C. 0.433

D. 0.533

Solution: Review Solution for Number 7

Problem 8: ECE Board April 1997

Evaluate the integral of sin6 x dx from 0 to π/2.

A. π/32

B. 2π/17

C. 3π/32

D. 5π/32

Solution: Review Solution for Number 8

Problem 9: CE Board May 1997

Evaluate the integral of x(x – 5)12 dx from 5 to 6.

A. 0.456

B. 0.556

C. 0.656

D. 0.756

Solution: Review Solution for Number 9

Problem 10: CE Board November 1996

Evaluate the integral of ((x dx) / (x + 1)8) from 0 to 1.

A. 0.011

B. 0.022

C. 0.033

D. 0.044

Solution: Review Solution for Number 10

Problem 11: ECE Board April 1998

Evaluate the integral of (cos 3A)8 dA from 0 to π/6.

A. 27π/363

B. 35π/768

C. 23π/765

D. 12π/81

Solution: Review Solution for Number 11

Problem 12: EE Board March 1998

Integrate (1 / (3x + 4)) with respect to x and evaluate the result from x = 0 and x = 2.

A. 0.278

B. 0.336

C. 0.252

D. 0.305

Solution: Review Solution for Number 12

Problem 13: ECE Board November 1991

Evaluate the integral of cos2 ydy.

A. (y/2) + (sin 2y/4) + C

B. y + 2cos y + C

C. (y/4) + (sin 2y/4) + C

D. y + sin 2y + C

Solution: Review Solution for Number 13

Problem 14: ECE Board November 1998

Integrate the square root of (1 – cos x) dx.

A. -2√2 cos (x/2) + C

B. -2√2 cos x + C

C. 2√2 cos (x/2) + C

D. 2√2 cos x + C

Solution: Review Solution for Number 14

Problem 15: ME Board October 1997

Evaluate the integral of cos x dx limits from π/4 to π/2.

A. 0.423

B. 0.293

C. 0.923

D. 0.329

Solution: Review Solution for Number 15

Problem 16: EE Board April 1997

Evaluate the integral of ln x dx, the limit are 1 and e.

A. 0

B. 1

C. 2

D. 3

Solution: Review Solution for Number 16

Problem 17: EE Board October 1997

Evaluate the integral of (2log10 e dx) / x from 1 to 10.

A. 2.0

B. 49.7

C. 3.0

D. 5.12

Solution: Review Solution for Number 17

Problem 18: CE Board May 1995

What is the integral of cos 2x esin 2x dx?

A. (esin 2x / 2) + C

B. –( esin 2x / 2) + C

C. – esin 2x + C

D. esin 2x + C

Solution: Review Solution for Number 18

Problem 19: ME Board April 1995, ME Board October 1997

The integral of cos x with respect to x is

A. sin x + C

B. sec x + C

C. –sin x + C

D. csc x + C

Solution: Review Solution for Number 19

Problem 20: EE Board April 1997

Find the integral of [(ex – 1) / (ex + 1)] dx

A. ln (ex – 1)2 + x + C

B. ln (ex + 1) – x + C

C. ln (ex – 1) + x + C

D. ln (ex + 1)2 – x + C

Solution: Review Solution for Number 20

Problem 21: EE Board April 1997

Evaluate the double integral of r sin u dr du, the limits of r is 0 and cos u and the limit of u are 0 and pi.

A. 1

B. 1/2

C. 0

D. 1/3

Solution: Review Solution for Number 21

Problem 22: CE Board November 1996

Evaluate the integral of (3x2 + 9y2) dx dy if the interior limits has an upper limit of y and a lower limit of 0, and whose outer limit has an upper limit of 2 and a lower limit of 0.

A. 10

B. 20

C. 30

D. 40

Solution: Review Solution for Number 22

Problem 23: EE Board April 1996

Evaluate the integral: A. 2/3

B. 4/3

C. 1/3

D. 5/3

Solution: Review Solution for Number 23

Problem 24: EE Board April 1993

Find the area of the region bounded by y2 = 8x and y = 2x.

A. 1.22 sq. units

B. 1.33 sq. units

C. 1.44 sq. units

D. 1.55 sq. units

Solution: Review Solution for Number 24

Problem 25: CE Board November 1994

What is the area bounded by the curve x2 = -9y and the line y + 1 = 0?

A. 3 sq. units

B. 4 sq. units

C. 5 sq. units

D. 6 sq. units

Solution: Review Solution for Number 25

Problem 26: CE Board May 1995

What is the area (in square units) bounded by the curve y2 = x and the line x – 4 = 0?

A. 30/3 sq. units

B. 31/3 sq. units

C. 32/3 sq. units

D. 29/3 sq. units

Solution: Review Solution for Number 26

Problem 27: EE Board October 1997

Find the area bounded by the curve y = x2 + 2 and the lines x = 0 and y = 0 and x = 4.

A. 88/3 sq. units

B. 64/3 sq. units

C. 54/3 sq. units

D. 64/5 sq. units

Solution: Review Solution for Number 27

Problem 28: EE Board April 1997

Find the area bounded by the parabolas y = 6x – x2 and y = x2 – 2x. Note. The parabolas intersect at points (0, 0) and (4, 8).

A. 44/3 sq. units

B. 64/3 sq. units

C. 74/3 sq. units

D. 54/3 sq. units

Solution: Review Solution for Number 28

Problem 29: ME Board April 1999

Find the area bounded by the parabola x2 = 4y and y = 4.

A. 21.33 sq. units

B. 33.21 sq. units

C. 31.32 sq. units

D. 13.23 sq. units

Solution: Review Solution for Number 29

Problem 30: EE Board October 1997

Find the area bounded by the line x – 2y + 10 = 0, the x-axis, the y-axis and x = 10.

A. 75 sq. units

B. 50 sq. units

C. 100 sq. units

D. 25 sq. units

Solution: Review Solution for Number 30

Problem 31: CE Board May 1996

What is the area (in square units) bounded by the curve y2 = 4x and x2 = 4y?

A. 5.33 sq. units

B. 6.67 sq. units

C. 7.33 sq. units

D. 8.67 sq. units

Solution: Review Solution for Number 31

Problem 32: CE Board May 1997

Find the area enclosed by the curve x2 + 8y + 16 = 0, the x-axis, the y-axis and the line x – 0.

A. 7.67 sq. units

B. 8.67 sq. units

C. 9.67 sq. units

D. 10.67 sq. units

Solution: Review Solution for Number 32

Problem 33: ME Board October 1997

What is the area bounded by the curve y = x3, the x-axis, and the line x = -2 and x = 1?

A. 4.25 sq. units

B. 2.45 sq. units

C. 5.24 sq. units

D. 5.42 sq. units

Solution: Review Solution for Number 33

Problem 34: ME Board April 1999

Find the area in the first quadrant bounded by the parabola y2 = 4x, x = 1, and x = 3.

A. 9.555 sq. units

B. 9.955 sq. units

C. 5.955 sq. units

D. 5.595 sq. units

Solution: Review Solution for Number 34

Problem 35: ECE Board April 1998

Find the area (in sq. units) bounded by the parabolas x2 – 2y = 0 and x2 + 2y – 8 = 0.

A. 11.77 sq. units

B. 4.7 sq. units

C. 9.7 sq. units

D. 10.7 sq. units

Solution: Review Solution for Number 35

Problem 36: ME Board April 1998

What is the area between y = 0, y = 3x2, x = 0, x – 2?

A. 8 sq. units

B. 24 sq. units

C. 12 sq. units

D. 6 sq. units

Solution: Review Solution for Number 36

Problem 37: CE Board May 1995

What is the area bounded by the curve y2 = x and the line x – 4 = 0?

A. 11 sq. units

B. 31/3 sq. units

C. 10 sq. units

D. 32/3 sq. units

Solution: Review Solution for Number 37

Problem 38: CE Board November 1996, CE Board November 1998

Find the area of the curve r2 = a2cos 2θ.

A. a sq. units

B. 22 sq. units

C. a2 sq. units

D. a3 sq. units

Solution: Review Solution for Number 38

Problem 39:

Locate the centroid of the plane area bounded by y = x2 and y = x.

A. 0.4 from the x-axis and 0.5 from the y-axis

B. 0.5 from the x-axis and 0.4 from the y-axis

C. 0.5 from the x-axis and 0.5 from the y-axis

D. 0.4 from the x-axis and 0.4 from the y-axis

Solution: Review Solution for Number 39

Problem 40:

Find the coordinates of the centroid of the plane area bounded by the parabola y = 4 – x2 and the x-axis.

A. (0, 1)

B. (0, 1.6)

C. (0, 2)

D. (1, 0)

Solution: Review Solution for Number 40

Problem 41:

Locate the centroid of the plane area bounded by the equation y2 = 4x, x = 1 and the x-axis on the first quadrant.

A. (3/4, 3/5)

B. (3/5, 3/4)

C. (2/3, 3/5)

D. (3/5, 2/3)

Solution: Review Solution for Number 41

Problem 42:

Find the length of the arc of the parabola x2 = 4y from x = -2 to x = 2.

A. 4.2 units

B. 4.6 units

C. 4.9 units

D. 5.2 units

Solution: Review Solution for Number 42

Problem 43:

Find the surface area (in square units) generated by rotating the parabola arc y = x2 about the x-axis from x = 0 to x = 1.

A. 5.33

B. 4.98

C. 5.73

D. 4.73

Solution: Review Solution for Number 43

Problem 44: CE Board May 1997

The area enclosed by the ellipse (x2/9) + (y2/4) = 1 is revolved about the line x = 3. What is the volume generated?

A. 355.3 cubic units

B. 360.1 cubic units

C. 370.3 cubic units

D. 365.1 cubic units

Solution: Review Solution for Number 44

Problem 45: CE Board May 1996

The area in the second quadrant of the circle x2 + y2 = 36 is revolved about line y + 10 = 0. What is the volume generated?

A. 2218.33 cubic units

B. 2228.83 cubic units

C. 2233.43 cubic units

D. 2208.53 cubic units

Solution: Review Solution for Number 45

Problem 46: CE Board November 1995

The area bounded by the curve y2 = 12x and the line x = 3 is revolved about the line x = 3. What is the volume generated?

A. 179 cubic units

B. 181 cubic units

C. 183 cubic units

D. 185 cubic units

Solution: Review Solution for Number 46

Problem 47: CE Board November 1994

Given the area in the first quadrant bounded by x2 = 8y, the line y – 2 = 0 and the y-axis. What is the volume generated when the area is revolved about the line y – 2 = 0?

A. 28.41 cubic units

B. 27.32 cubic units

C. 25.83 cubic units

D. 26.81 cubic units

Solution: Review Solution for Number 47

Problem 48:

Find the volume (in cubic units) generated by rotating a circle x2 + y2 + 6x + 4y + 12 = 0 about the y-axis.

A. 39.48 cubic units

B. 47.23 cubic units

C. 59.22 cubic units

D. 62.11 cubic units

Solution: Review Solution for Number 48

Problem 49: CE Board May 1995

Given the area in the first quadrant by x2 = 8y, the line x = 4 and the x-axis. What is the volume generated by revolving this area about the y-axis.

A. 53.26 cubic units

B. 52.26 cubic units

C. 51.26 cubic units

D. 50.26 cubic units

Solution: Review Solution for Number 49

Problem 50: CE Board November 1995

Find the moment of inertia with respect to x-axis of the area bounded by the parabola y2 = 4x and the line x = 1.

A. 2.03

B. 2.13

C. 2.33

D. 2.53

Solution: Review Solution for Number 50

### Online Question and Answer in Integral Calculus Series

Following is the list of multiple choice questions in this brand new series:

MCQ in Integral Calculus
PART 1: MCQ from Number 1 – 50                                       Answer key: PART 1
PART 2: MCQ from Number 51 – 100                                   Answer key: PART 2