Calculus 2 Online Quizzes: Master Integration Techniques and Applications

Question 1

To evaluate the integral ∫(x^3 * sin(x)) dx, which integration technique is most suitable?

Accepted Answer

Integration by parts

Answer

Substitution

Answer

Trigonometric substitution

Answer

Partial fractions decomposition

Question 2

Which of the following is not a method of integration?

Accepted Answer

Differentiation

Answer

Integration by parts

Answer

Integration by substitution

Answer

Integration by partial fractions

Question 3

Evaluate the integral ∫(cos²(x) + sin(x)cos(x)) dx using an appropriate trigonometric identity.

Accepted Answer

1/2 sin(2x) + C

Answer

sin(x) + cos(x) + C

Answer

-sin(x) + cos(x) + C

Answer

-1/2 sin(2x) + C

Question 4

Which of the following integrals is considered improper?

Accepted Answer

∫[-∞, ∞] f(x) dx

Accepted Answer

∫[0, 1] f(x) dx, where f(x) is discontinuous at x = 0.5

Answer

∫[0, 1] f(x) dx

Question 5

Find the derivative of the function f(x) = x^2 * e^(2x) with respect to x.

Accepted Answer

2x * e^(2x) + x^2 * 2e^(2x)

Answer

2x * e^(2x) + 2x^2 * e^(2x)

Answer

2x * e^(2x)

Answer

x^2 * e^(2x)

Question 6

Which of the following is a mathematical operation that is NOT a fundamental technique of integration?

Accepted Answer

Differentiation

Answer

Integration by substitution

Answer

Integration by parts

Question 7

Find the integral of ∫(x² + 2x + 1) dx

Accepted Answer

x³/3 + x² + x + C

Answer

x³ + x² + 1 + C

Answer

1/3x³ + x² + x + C

Answer

x³ + 2x² + x + C

Question 8

Which of the following is NOT a property of definite integrals

Accepted Answer

∫[a, b] f(x) dx = ∫[b, a] f(x) dx

Answer

∫(f(x) + g(x)) dx = ∫f(x) dx + ∫g(x) dx

Answer

∫c f(x) dx = c ∫f(x) dx

Answer

∫[a, c] f(x) dx = ∫[a, b] f(x) dx + ∫[b, c] f(x) dx

Question 9

Which of the following is a method for integrating rational functions?

Accepted Answer

Partial Fraction Decomposition

Answer

Integration by Substitution

Answer

Integration by Parts

Answer

Trig Substitution

Question 10

Determine the volume of the solid generated by rotating the region bounded by y = √x, y = 0, and x = 4 about the x-axis.

Accepted Answer

8π/5

Answer

2π/5

Answer

4π/5

Answer

16π/5

Question 11

What is the formula for the work done by a constant force F over a displacement d?

Accepted Answer

W = F * d

Answer

W = 1/2 * F * d

Answer

W = F * d^2

Answer

W = 1/2 * F * d^2

Question 12

Determine the surface area of the surface generated by rotating the curve y = x^3 from x = 0 to x = 1 about the y-axis.

Accepted Answer

2π/5

Answer

π/5

Answer

π

Answer

2π

Question 13

Find the work done by the force F(x) = 2x + 3 applied over the interval [0, 2].

Accepted Answer

14

Answer

7

Answer

21

Answer

28

Question 14

Evaluate the indefinite integral: ∫cos(3x) dx.

Accepted Answer

sin(3x) / 3 + C

Answer

sin(x) / 3 + C

Answer

3sin(3x) + C

Answer

1/3 cos(3x) + C

Question 15

Which of the following is the definite integral representation of the area under the curve y = f(x) from a to b?

Accepted Answer

∫[a, b] f(x) dx

Answer

∫[0, a] f(x) dx

Answer

∫[b, a] f(x) dx

Answer

∫[0, b] f(x) dx

Question 16

Find the volume of the solid generated by rotating the region bounded by y = x and y = x^2 about the y-axis.

Accepted Answer

π/6

Answer

π/12

Answer

π/3

Answer

π/2

Question 17

Which of the following techniques can be used to evaluate the integral: ∫x^2 sin(x) dx?

Accepted Answer

Integration by parts

Answer

Substitution

Answer

Trig substitution

Answer

U-substitution

Question 18

Which of the following is the correct formula for the arc length of the curve y = f(x) from a to b?

Accepted Answer

∫[a, b] √(1 + (dy/dx)^2) dx

Answer

∫[a, b] √(1 - (dy/dx)^2) dx

Answer

∫[a, b] (dy/dx) dx

Answer

∫[a, b] f(x) dx

Question 19

Determine the convergence of the following improper integral: ∫[0,∞) (x^2 + 1)/(x^3 + 2x) dx

Accepted Answer

Converges to 1/2

Answer

Converges to 1

Answer

Diverges to infinity

Answer

Diverges and does not converge

Question 20

Find the area of the region bounded by the curve y = x^3 and the horizontal line y = 8.

Accepted Answer

16

Answer

4

Answer

8

Answer

64

Question 21

Calculate the volume of the solid obtained by rotating the region bounded by the curve y = sqrt(x) and the line y = x about the x-axis.

Accepted Answer

π/3 cubic units

Answer

π/6 cubic units

Answer

π cubic units

Answer

2π cubic units

Question 22

Use integration by parts to evaluate the integral: ∫x*e^x dx

Accepted Answer

x*e^x - e^x + C

Answer

xe^x + C

Answer

e^x - x*e^x + C

Answer

e^x + C

Question 23

Which of the following is an appropriate substitution for the integral: ∫(x^2 + 1)^3 dx?

Accepted Answer

u = x^2 + 1

Answer

u = x + 1

Answer

u = x^3 + 1

Answer

u = x^2 + 2x + 1

Question 24

Determine the convergence of the infinite series: ∑[n=1,∞] (n/(n^2 + 1))

Accepted Answer

Converges

Answer

Diverges

Answer

Cannot be determined

Answer

Converges absolutely

Question 25

Given a demand function p = 100 - 2q, where p represents the price and q represents the quantity, determine the consumer surplus using a definite integral.

Accepted Answer

∫(100 - 2q) dq

Answer

∫(p) dq

Answer

∫(100 + 2q) dq

Answer

∫(2q - 100) dq

Question 26

Find the integral of the function f(x) = x^3 + 2x^2 - 5 with respect to x.

Accepted Answer

x^4/4 + 2x^3/3 - 5x + C

Answer

x^4 + 2x^3 - 5x + C

Answer

x^3/4 + 2x^2/3 - 5x + C

Answer

x^2/4 + 2x^3 - 5x + C

Question 27

Which of the following integrals represents the area under the curve y = sin(x) over the interval [0, π]?

Accepted Answer

∫[0,π] sin(x) dx

Answer

∫[π,0] sin(x) dx

Answer

∫[0,π] cos(x) dx

Answer

∫[π,0] cos(x) dx

Question 28

Find the integral of the rational function f(x) = x/(x^2 + 1) with respect to x.

Accepted Answer

1/2 ln(x^2 + 1) + C

Answer

ln(x^2 + 1) + C

Answer

1/2 arctan(x) + C

Answer

arctan(x) + C

Question 29

Find the integral of the function f(x) = x^2/(1 + x^2) with respect to x.

Accepted Answer

x - arctan(x) + C

Answer

x + arctan(x) + C

Answer

1/2 ln(1 + x^2) + C

Answer

1/2 arctan(x) + C

Question 30

Evaluate the integral ∫(x² + sin(x)) dx.

Accepted Answer

(x³/3) + cos(x) + C

Answer

(x³/3) - cos(x) + C

Answer

x² - sin(x) + C

Answer

x² + cos(x) + C

Question 31

Find the area of the region bounded by the curves y = x² and y = 4.

Accepted Answer

16/3

Answer

8/3

Answer

32/3

Answer

64/3

Question 32

Determine which of the following integrals is improper:

Accepted Answer

∫[0,∞] 1/x^2 dx

Answer

∫[0,π] cos(x)dx

Answer

∫[0,1] sin(x)dx

Answer

∫[-1,1] |x| dx

Question 33

Which of the following methods is most suitable for evaluating integrals of the form ∫(a + bx)^n dx?

Accepted Answer

u-substitution

Answer

trigonometric substitution

Answer

partial fractions

Answer

integration by parts

Question 34

Which formula represents the area under the curve y = f(x) from a to b?

Accepted Answer

∫[a, b] f(x) dx

Answer

∫[b, a] f(x) dx

Answer

f(x) dx

Answer

∫[a, b] x f(x) dx

Question 35

Calculate the arc length of the curve y = x^3/3 from x = 0 to x = 2.

Accepted Answer

∫[0, 2] √(1 + (d/dx)(x^3/3)^2) dx

Answer

∫[0, 2] √(1 + x^2) dx

Answer

∫[0, 2] x^2/3 dx

Answer

∫[0, 2] x^3/3 dx

Question 36

Determine the work done by the force F(x) = x^2 + 2x from x = 0 to x = 3.

Accepted Answer

∫[0, 3] (x^2 + 2x) dx

Answer

∫[0, 3] x^2 dx

Answer

∫[0, 3] x^3 + x^2 dx

Answer

F(x) dx

Question 37

Evaluate the integral ∫(x^2 + sin(x)) dx.

Accepted Answer

x^3/3 - cos(x) + C

Answer

x^2/2 + sin(x) + C

Answer

x^2 - cos(x) + C

Answer

x^3/2 - sin(x) + C

Question 38

Evaluate the definite integral ∫[0, π/2] sin(x) dx.

Accepted Answer

1

Answer

π

Answer

0

Answer

-1

Question 39

Determine the volume of the solid generated by rotating the region bounded by y = x^2 and y = 4 about the x-axis.

Accepted Answer

128π/15

Answer

256π/15

Answer

128π/5

Answer

64π/15

Question 40

Find the work done by the force field F(x, y) = (x^2, y^3) in moving an object along the curve C: r(t) = (t, t^2) from t = 0 to t = 1.

Accepted Answer

1/3

Answer

1/2

Answer

1

Question 41

Find the area of the surface generated by rotating the curve y = √x from x = 0 to x = 4 about the x-axis.

Accepted Answer

(8π√2) / 3

Answer

(4π√2) / 3

Answer

(16π√2) / 3

Answer

(8π) / 3

Question 42

Use the divergence theorem to calculate the outward flux of the vector field F(x, y, z) = (x^2, y^2, z^2) across the surface of the sphere x^2 + y^2 + z^2 = 4.

Accepted Answer

32π

Answer

16π

Answer

0

Answer

64π

Question 43

Evaluate the integral: ∫(x^2 + 2x - 1) dx

Accepted Answer

x^3/3 + x^2 - x + C

Answer

x^2 + 2x - 1 + C

Answer

x^3 + 2x^2 - x + C

Answer

x^3/2 + 2x^2 - x + C

Question 44

Find the area under the curve y = sin(x) from x = 0 to x = π/2.

Accepted Answer

1

Answer

2π

Answer

π

Answer

π/2

Question 45

Use the method of integration by parts to evaluate the integral: ∫x ln(x) dx

Accepted Answer

x ln(x) - x + C

Answer

x^2 ln(x) + x + C

Answer

ln(x) + C

Answer

x^2 ln(x) - x + C

Question 46

Find the volume of the solid generated by rotating the region bounded by the curves y = x^2 and y = 4 about the x-axis.

Accepted Answer

64π/5

Answer

32π/5

Answer

64π/3

Answer

16π/5

Question 47

Evaluate the improper integral: ∫dx/(x^2 + 1)

Accepted Answer

arctan(x) + C

Answer

x^2 + C

Answer

1/(x^2 + 1) + C

Answer

ln(x^2 + 1) + C

Question 48

Find the convergence or divergence of the following series: ∑(n=1 to ∞) (n^2 + 1)/(n^3 - 1)

Accepted Answer

Converges

Answer

Conditional convergence

Answer

Diverges

Answer

Cannot be determined

Question 49

Determine whether the following function is continuous at x = 0: f(x) = |x - 1|

Accepted Answer

Not continuous

Answer

Continuous at x = 0 only

Answer

Cannot be determined

Answer

Continuous

Question 50

Find the derivative of the following function: f(x) = e^(x^2) sin(x)

Accepted Answer

2xe^(x^2) sin(x) + e^(x^2) cos(x)

Answer

x sin(x) + cos(x)

Answer

(2x^2) sin(x) + (2x) cos(x)

Answer

e^(x^2) sin(x) + e^(x^2) cos(x)

Question 51

Find the length of the curve given by the parametric equations: x = t^2, y = t^3 from t = 0 to t = 1.

Accepted Answer

√(10)/6

Answer

1/3

Answer

√(10)/3

Answer

1/6

Question 52

Evaluate the improper integral: ∫[x^(2) + 1 / x] dx from 1 to infinity

Accepted Answer

∞

Answer

0

Answer

-1

Answer

1

Question 53

Find the volume of the solid generated by rotating the region bounded by the curve y = x^2 and the lines y = 2 and x = 0 about the x-axis

Accepted Answer

8π/15

Answer

4π/5

Answer

4π/15

Answer

8π/5

Question 54

Use the method of substitution to evaluate the integral: ∫[sin(x^2) * cos(x^2)] * 2x dx

Accepted Answer

-cos(x^2) + C

Answer

cos(x^2) + C

Answer

sin(x^2) + C

Answer

-sin(x^2) + C

Question 55

Use polar coordinates to evaluate the integral: ∫∫[x^2 + y^2] dA over the region bounded by the circle x^2 + y^2 = 4

Accepted Answer

8π

Answer

4π

Answer

16π

Answer

32π

Question 56

Find the area of the surface generated by rotating the curve y = √x from x = 0 to x = 4 about the x-axis

Accepted Answer

16π

Answer

32π

Answer

8π

Answer

64π

Question 57

Which integration technique is most suitable for evaluating the integral of  ∫(x² + 1)e^x dx?

Accepted Answer

Integration by Parts

Answer

Trigonometric Substitution

Answer

Partial Fractions

Answer

U-Substitution

Question 58

What is the volume of the solid generated by revolving the region bounded by y = x², y = 0, x = 1, and x = 2 about the x-axis?

Accepted Answer

15π/2

Answer

31π/5

Answer

8π/3

Answer

7π

Question 59

Evaluate the improper integral: ∫(1/x) dx from 1 to infinity.

Accepted Answer

The integral diverges.

Answer

0

Answer

1

Answer

ln(2)

Question 60

What is the arc length of the curve y = (2/3)x^(3/2) from x = 0 to x = 3?

Accepted Answer

14/3

Answer

4√2

Answer

10/3

Answer

6

Question 61

Find the area of the region enclosed by the curves y = x² and y = 2x - x².

Accepted Answer

1/3

Answer

1/2

Answer

2/3

Answer

1

Question 62

Which of the following is the correct partial fraction decomposition for (2x + 1)/(x² + x - 2)?

Accepted Answer

1/(x - 1) + 3/(x + 2)

Answer

2/(x - 1) + 1/(x + 2)

Answer

(2x + 3)/(x² + x - 2)

Answer

(x + 1)/(x - 1) + (x - 1)/(x + 2)

Question 63

Determine the value of the integral: ∫cos²(x) dx from 0 to π/2.

Accepted Answer

π/4

Answer

π/2

Answer

π

Answer

2π

Question 64

A tank is filled with water at a rate of r(t) = 2t + 3 liters per minute. What is the total amount of water added to the tank during the first 4 minutes?

Accepted Answer

20 liters

Answer

26 liters

Answer

16 liters

Answer

32 liters