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

Answer: **Option A**

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

Answer: **Option D**

Solution: Review Solution for Number 2

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

Integrate xcos (2x^{2} + 7) dx.

A. (1/4)sin (2x^{2} + 7) + C

B. (1/4)cos (2x^{2} + 7) + C

C. ((sin θ) / 4(x^{2} + 7)) + C

D. sin (2x^{2} + 7) + C

Answer: **Option A**

Solution: Review Solution for Number 3

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

Integrate: (7x^{3} + 4x^{2}) dx

A. (7x^{3}/3) + (4x^{2}/2) + C

B. (7x^{4}/4) + (4x^{2}/5) + C

C. (7x^{4}/4) + (4x^{3}/3) + C

D. 7x^{4} + (4x^{2}/2) + C

Answer: **Option C**

Solution: Review Solution for Number 4

**Problem 5: CE Board November 1995**

What is the integral of sin^{5} x cos^{3} 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

Answer: **Option C**

Solution: Review Solution for Number 5

**Problem 6: CE Board November 1994**

What is the integral of sin^{5} 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

Answer: **Option D**

Solution: Review Solution for Number 6

**Problem 7: CE Board May 1996**

Find the integral of 12 sin^{5} x cos^{5} 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

Answer: **Option A**

Solution: Review Solution for Number 7

**Problem 8: ECE Board April 1997**

Evaluate the integral of sin^{6} x dx from 0 to π/2.

A. π/32

B. 2π/17

C. 3π/32

D. 5π/32

Answer: **Option D**

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

Answer: **Option A**

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

Answer: **Option B**

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

Answer: **Option B**

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

Answer: **Option D**

Solution: Review Solution for Number 12

**Problem 13: ECE Board November 1991**

Evaluate the integral of cos^{2} 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

Answer: **Option A**

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

Answer: **Option A**

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

Answer: **Option B**

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

Answer: **Option B**

Solution: Review Solution for Number 16

**Problem 17: EE Board October 1997**

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

A. 2.0

B. 49.7

C. 3.0

D. 5.12

Answer: **Option A**

Solution: Review Solution for Number 17

**Problem 18: CE Board May 1995**

What is the integral of cos 2x e^{sin 2x} dx?

A. (e^{sin 2x} / 2) + C

B. –( e^{sin 2x} / 2) + C

C. – e^{sin 2x} + C

D. e^{sin 2x} + C

Answer: **Option A**

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

Answer: **Option A**

Solution: Review Solution for Number 19

**Problem 20: EE Board April 1997**

Find the integral of [(e^{x} – 1) / (e^{x} + 1)] dx

A. ln (e^{x} – 1)^{2} + x + C

B. ln (e^{x} + 1) – x + C

C. ln (e^{x} – 1) + x + C

D. ln (e^{x} + 1)^{2} – x + C

Answer: **Option D**

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

Answer: **Option D**

Solution: Review Solution for Number 21

**Problem 22: CE Board November 1996**

Evaluate the integral of (3x^{2} + 9y^{2}) 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

Answer: **Option D**

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

Answer: **Option A**

Solution: Review Solution for Number 23

**Problem 24: EE Board April 1993**

Find the area of the region bounded by y^{2} = 8x and y = 2x.

A. 1.22 sq. units

B. 1.33 sq. units

C. 1.44 sq. units

D. 1.55 sq. units

Answer: **Option B**

Solution: Review Solution for Number 24

**Problem 25: CE Board November 1994**

What is the area bounded by the curve x^{2} = -9y and the line y + 1 = 0?

A. 3 sq. units

B. 4 sq. units

C. 5 sq. units

D. 6 sq. units

Answer: **Option B**

Solution: Review Solution for Number 25

**Problem 26: CE Board May 1995**

What is the area (in square units) bounded by the curve y^{2} = 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

Answer: **Option C**

Solution: Review Solution for Number 26

**Problem 27: EE Board October 1997**

Find the area bounded by the curve y = x^{2} + 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

Answer: **Option A**

Solution: Review Solution for Number 27

**Problem 28: EE Board April 1997**

Find the area bounded by the parabolas y = 6x – x^{2} and y = x^{2} – 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

Answer: **Option B**

Solution: Review Solution for Number 28

**Problem 29: ME Board April 1999**

Find the area bounded by the parabola x^{2} = 4y and y = 4.

A. 21.33 sq. units

B. 33.21 sq. units

C. 31.32 sq. units

D. 13.23 sq. units

Answer: **Option A**

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

Answer: **Option A**

Solution: Review Solution for Number 30

**Problem 31: CE Board May 1996**

What is the area (in square units) bounded by the curve y^{2} = 4x and x^{2} = 4y?

A. 5.33 sq. units

B. 6.67 sq. units

C. 7.33 sq. units

D. 8.67 sq. units

Answer: **Option A**

Solution: Review Solution for Number 31

**Problem 32: CE Board May 1997**

Find the area enclosed by the curve x^{2} + 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

Answer: **Option D**

Solution: Review Solution for Number 32

**Problem 33: ME Board October 1997**

What is the area bounded by the curve y = x^{3}, 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

Answer: **Option A**

Solution: Review Solution for Number 33

**Problem 34: ME Board April 1999**

Find the area in the first quadrant bounded by the parabola y^{2} = 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

Answer: **Option D**

Solution: Review Solution for Number 34

**Problem 35: ECE Board April 1998**

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

A. 11.77 sq. units

B. 4.7 sq. units

C. 9.7 sq. units

D. 10.7 sq. units

Answer: **Option D**

Solution: Review Solution for Number 35

**Problem 36: ME Board April 1998**

What is the area between y = 0, y = 3x^{2}, x = 0, x – 2?

A. 8 sq. units

B. 24 sq. units

C. 12 sq. units

D. 6 sq. units

Answer: **Option A**

Solution: Review Solution for Number 36

**Problem 37: CE Board May 1995**

What is the area bounded by the curve y^{2} = 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

Answer: **Option D**

Solution: Review Solution for Number 37

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

Find the area of the curve r^{2} = a^{2}cos 2θ.

A. a sq. units

B. 2^{2} sq. units

C. a^{2} sq. units

D. a^{3} sq. units

Answer: **Option C**

Solution: Review Solution for Number 38

**Problem 39:**

Locate the centroid of the plane area bounded by y = x^{2} 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

Answer: **Option B**

Solution: Review Solution for Number 39

**Problem 40:**

Find the coordinates of the centroid of the plane area bounded by the parabola y = 4 – x^{2} and the x-axis.

A. (0, 1)

B. (0, 1.6)

C. (0, 2)

D. (1, 0)

Answer: **Option B**

Solution: Review Solution for Number 40

**Problem 41:**

Locate the centroid of the plane area bounded by the equation y^{2} = 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)

Answer: **Option B**

Solution: Review Solution for Number 41

**Problem 42:**

Find the length of the arc of the parabola x^{2} = 4y from x = -2 to x = 2.

A. 4.2 units

B. 4.6 units

C. 4.9 units

D. 5.2 units

Answer: **Option B**

Solution: Review Solution for Number 42

**Problem 43:**

Find the surface area (in square units) generated by rotating the parabola arc y = x^{2} about the x-axis from x = 0 to x = 1.

A. 5.33

B. 4.98

C. 5.73

D. 4.73

Answer: **Option A**

Solution: Review Solution for Number 43

**Problem 44: CE Board May 1997**

The area enclosed by the ellipse (x^{2}/9) + (y^{2}/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

Answer: **Option A**

Solution: Review Solution for Number 44

**Problem 45: CE Board May 1996**

The area in the second quadrant of the circle x^{2} + y^{2} = 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

Answer: **Option B**

Solution: Review Solution for Number 45

**Problem 46: CE Board November 1995**

The area bounded by the curve y^{2} = 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

Answer: **Option B**

Solution: Review Solution for Number 46

**Problem 47: CE Board November 1994**

Given the area in the first quadrant bounded by x^{2} = 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

Answer: **Option D**

Solution: Review Solution for Number 47

**Problem 48: **

Find the volume (in cubic units) generated by rotating a circle x^{2} + y^{2} + 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

Answer: **Option C**

Solution: Review Solution for Number 48

**Problem 49: CE Board May 1995**

Given the area in the first quadrant by x^{2} = 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

Answer: **Option D**

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 y^{2} = 4x and the line x = 1.

A. 2.03

B. 2.13

C. 2.33

D. 2.53

Answer: **Option B**

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**

**MCQ from Number 1 – 50**Answer key:

**PART 1**

**MCQ from Number 51 – 100**Answer key:

**PART 2**

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