Mechanical Engineering Review Online

1. Calculate the definite integral of ∫(5x3 - 4x2)dx upper limit: 2; lower limit: -2

a. -19.25

b. -21.33

c. -23.51

d. -25.44

2. Find the integral of ∫x/(x²+2)dx 

a. 0.5ln(x² + 2) + C

b. ln(x² + 1) + C

c. -ln(x² + 2) + C

d. 2ln(x2 + 1) + C

3. Find the integral of ∫(2-x)^0.5dx

a. 2/3(2-x)^1.5+C

b. 2/3(2-x)+C

c. 2/3(x-2)^1.5+C

d. 3/2(2-x)^1.5+C

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

a. 15.21

b. 16.67

c. 18.32

d. 12.21

5. What is the area bounded by y = 0, y = e^x, x =0 and x = 2?

a. 3.21

b. 7.85

c. 5.74

d. 6.40

6. What is the area bounded by the curve y = x³, the x-axis, and the lines x = -2 and x = 2?

a. 7

b. 6

c. 8

d. 10

7. Find the area bounded by y = x² and y = 3x² – 6x.

a. 11

b. 8

c. 9

d. 7

8. Find the area bounded by the x-axis, y² = x, x = 2 and x = 18.

a. 32

b. 46

c. 49

d. 37

9. Find the area bounded by y² = 10x and line x = y.

a. 19.25

b. 21.33

c. 24.81

d. 32.22 

10. Find the area between x² = -3y + 12 and x-axis.

a. 15.54

b. 16.38

c. 18.22

d. 18.45

11. Find the area between y² = 6x + 24 and x = 4.

a. 64.87

b. 55.24

c. 80.11

d. 73.92

12. Find the area bounded by the line 2x – 8y = 14, x-axis and y-axis.

a. 4.25

b. 8.20

c. 6.13

d. 7.14

13. Find the centroid of the area bounded by the following curves 3x + y = 6, x = 0, y = 0.

a. (-2/3, 3)

b. (2/3, 2)

c. (2/3, 1)

d. (-2/3, 0)

14. Find the volume generated by revolving about y-axis the area bounded by the following curves y = 2x² and y = 2x.

a. 1/3π

b. 1/4π

c. 3/4π

d. 1/2π

15. Find the volume generated by revolving about x-axis the area bounded by the following curves y = x³, y = 0 and x = 3.

a. 981.5

b. 732.0

c. 820.1

d. 650.3

16. Find the volume generated by revolving about x-axis the area bounded by the following curves y = x² and y = 3x.

a. 92.21

b. 101.8

c. 120.3

d. 132.1

17. Solve the differential equation 3ydx + (xy + 6x)dy = 0

a. x³y⁶e^y=C

b. 2x³y⁶e^y=C

c. -x³y⁶e^y=C

d. x⁶y³e^y=C

18. Solve the differential equation xdy + ydx = 0.

a. x/y = C

b. –xy=C

c. y/x=C

d. xy=C

19. Solve the differential equation of 3xy + 9x + (x2 – 5)y’ = 0

a. (x2-6)^1.5(y-3)=C

b. (x2+5)^1.5(y-3)=C

c. (x2-5)^1.5(y+3)=C

d. (x2+6)^1.5(y+3)=C

20. Find the particular solution of y’ = 3x which satisfies the condition that y = 8 if x = 2.

a. 2x²+2y+5=0

b. 3x²+2y-5=0

c. 3x²-2y+4=0

d. 5x²-2y+5=0

Answer Key (Handwritten)