Module 09: Integral Calculus and Differential Equations – Mechanical Engineering Online Review Center

Handouts
Exercises
Integration Formula and Area Under a Curve
Volume of Solid Revolution

LINK: Integral Calculus and Differential Equations

Topics:

The Power Rule
Integration by parts
Integration of Logarithmic functions
Integration of Trigonometric functions
Area bounded by a curve
Determination of volume by Integration
Centroid of a plane area
Differential Equations

Calculate the definite integral of ∫(5×3 – 4×2)dx upper limit: 2; lower limit: -2
- a. -19.25
- b. -21.33
- c. -23.51
- d. -25.44
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
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
Find the area bounded by the parabola y = x² and x = 4.
- a. 15.21
- b. 16.67
- c. 18.32
- d. 12.21
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
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
Find the area bounded by y = x² and y = 3x² – 6x.
- a. 11
- b. 8
- c. 9
- d. 7
Find the area bounded by the x-axis, y² = x, x = 2 and x = 18.
- a. 32
- b. 46
- c. 49
- d. 37
Find the area bounded by y² = 10x and line x = y.
- a. 19.25
- b. 21.33
- c. 24.81
- d. 32.22
Find the area between x² = -3y + 12 and x-axis.
- a. 15.54
- b. 16.38
- c. 18.22
- d. 18.45
Find the area between y² = 6x + 24 and x = 4.
- a. 64.87
- b. 55.24
- c. 80.11
- d. 73.92
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
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)
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
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
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
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
Solve the differential equation xdy + ydx = 0.
- a. x/y = C
- b. –xy=C
- c. y/x=C
- d. xy=C
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
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)