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.

### Multiple Choice Questions Topic Outline

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

### Online Questions and Answers in Integral Calculus Series

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

**Integral Calculus MCQs**

**MCQs from Number 1 – 50**Answer key:

**PART I**

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

**PART II**

### Start Practice Exam Test Questions Part I 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

**Problem 2: ECE Board November 1998**

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

- A. 2
^{1/2} - B. 1/2
- C. ln 3
- D. ln 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 Î¸) / 4(x
^{2}+ 7)) + C - D. sin (2x
^{2}+ 7) + C

**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

**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 Ï€/2?

- A. 0.0203
- B. 0.0307
- C. 0.0417
- D. 0.0543

**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 Ï€/2?

- A. 0.233
- B. 0.333
- C. 0.433
- D. 0.533

**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 Ï€/2.

- A. 0.2
- B. 0.333
- C. 0.433
- D. 0.533

**Problem 8: ECE Board April 1997**

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

- A. Ï€/32
- B. 2Ï€/17
- C. 3Ï€/32
- D. 5Ï€/32

**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

**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

**Problem 11: ECE Board April 1998**

Evaluate the integral of (cos 3A)^{8} dA from 0 to Ï€/6.

- A. 27Ï€/363
- B. 35Ï€/768
- C. 23Ï€/765
- D. 12Ï€/81

**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

**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

**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

**Problem 15: ME Board October 1997**

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

- A. 0.423
- B. 0.293
- C. 0.923
- D. 0.329

**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

**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

**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

**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

**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

**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

**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

**Problem 23: EE Board April 1996**

Evaluate the integral:

- A. 2/3
- B. 4/3
- C. 1/3
- D. 5/3

**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

**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

**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

**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

**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

**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

**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

**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

**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

**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

**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

**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

**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

**Problem 37: CE Board May 1995**

- A. 11 sq. units
- B. 31/3 sq. units
- C. 10 sq. units
- D. 32/3 sq. units

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

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

- A. a sq. units
- B. 2a sq. units
- C. a
^{2}sq. units - D. a
^{3}sq. units

**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

**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)

**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)

**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

**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

**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

**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

**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

**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

**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

**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

**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

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