Integral Calculus

• Integrate: ∫ x^2 e^(3x) dx

  A. (1/3)x^2e^3x – (2/9)xe^3x + (2/27)e^3x + C

  B. (2/3)x^2e^3x – (1/3)xe^3x + (1/27e^3x + C

  C. (2/9)x^2e^3x – (2/3)xe^3x + (5/27)e^3x + C

  D. (2/3)x^2e^3x – (1/9)xe^3x + (2/27)e^3x + C
• Determine the area of the region bounded by the parabola y=9 – x 2 and the line x + y = 7.

  A. 9/2

  B. 3/2

  C. 5/2

  D. 7/2
• Find the volume of the solid revolution obtained by revolving the region bounded by y=x-x^2 and the x axis about the x axis.

  A. π/15

  B. π/30

  C. π/45

  D. π/60
• Find the volume obtained if the region bounded by y = x^2 and y = 2x is rotated about the x-axis.

  A. 34π/15

  B. 54π/5

  C. 64π/15

  D. 14π/5
• Determine the area of the region bounded by the curve y = x^3 – 4x^2 + 3x and the x axis, 0 ≤ x ≤ 3.

  A. 9/4

  B. 37/12

  C. -9/4

  D. 17/3
• Determine the area of the region bounded by the curves y = x^4 – x^2 and y = x^2 – 1.

  A. 16/13

  C. 15/4

  B. 16/15

  D. 17/3
• Find the area of the region bounded by the parabola y = x^2 , the tangent line to the parabola at the point (2, 4), and the x axis.

  A. 1/3

  B. 1

  C. 2/3

  D. 4/3
• A hole of radius 2 is drilled through the axis of a sphere of radius 3. Compute the volume of the remaining solid.

  A. 46.832

  B. 38.234

  C. 35.235

  D. 50.234
• Determine the area bounded by the curve y^2= 9x/5 and the line y= x - 2.

  A. 5.27

  B. 8.25

  C. 7.59

  D. 6.86
• . Find the distance of the centroid from the y-axis of the area bounded by the curve x^2=16y, the line x=12 and the x-axis.

  A. 8

  B. 4

  C. 9

  D. 3
• 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. 4.12

  B. 3.16

  C. 2.13

  D. 5.18
• . The area on the first and second quadrant of the circle x^2+y^2=36 is revolved about the line y=6. What is the volume generated?

  A. 1235.80 cu. units

  B. 1225.80 cu. units

  C. 1245.80 cu. units

  D. 1325.80 cu. units
• A body moves along a straight path such that its velocity is given by the expression v = 2 +1/2t + 1/3 t^2 where v is in m/s and t is in sec. If the distance traveled after 1 sec. is 2.361 m., find the distance it travels at the end of 3 sec. A. 23.67 m

  C. 21.50 m

  B. 12.56 m

  D. 11.25 m
• . An object experiences rectilinear acceleration a(t)=10-2t . How far does it travel in 6 seconds if its initial velocity is 10m/s.

  A. 254 m

  B.168 m

  C. 287 m

  D. 133 m
• . A 5n lb. monkey is attached to a 20 ft. hanging rope that weighs 0.3 lb/ft. The monkey climbs the rope up to the top. How much work has it done?

  A. 160 ft.-lb.

  B. 445 ft.-lb.

  C. 325 ft.-lb.

  D. 232 ft.-lb
• A spring with a natural length of 10 cm, is stretched by 1/2 cm. by a 12 Newton force. Find the work done in stretching the spring from 10 cm. to 18 cm. Express your answer in joules.

  A. 6.68 Joules

  B. 14.68 Joules

  C. 7.68 Joules

  D. 10.68 Joules
• Evaluate the following integral ∫ 0 TO 2 (3^(2X) dx

  A. 36.41

  B. 28.67

  C. 45.73

  D. 24.58
• Evaluate ∫ tan 𝜃 ln sec 𝜃 𝑑𝜃.

  a. 2(ln sec 𝜃) 2 + 𝐶

  b. (ln sec 𝜃) 2 + 𝐶

  c. ½ (ln sec 𝜃) + 𝐶

  d. ½ (ln sec 𝜃) 2 + C
• Find the length of the arc of x^2 + y^2 =64 from x= -1 to x=-3, in the second quadrant.

  A. 3.15

  B. 2.07

  C. 3.22

  D. 2.16
• An object experiences rectilinear acceleration a(t)=10-2t . How far does it travel in 6 seconds if its initial velocity is 10m/s.

  A. 128 m

  B. 168 m

  C. 148 m

  D. 188 m
• Find ∫ 0 to 3 ∫ 0 to 2 (3𝑥 + 𝑥^2𝑦)𝑑𝑦𝑑𝑥

  A. 18

  B. 63

  C. 45

  D. 100
• Find ∫ 0 to 2 ∫ 0 to y (3𝑥^2 + 9𝑦^2 )𝑑𝑥𝑑𝑦

  A. 30

  B. 50

  C. 40

  D. 20