Applications of the Definite Integral: Master Area, Volume, and Work Calculations

Question 1

Find precisely the area under the curve y = e^(-x^2) from x = -∞ to x = ∞.

Accepted Answer

1

Answer

0

Answer

π

Answer

2π

Question 2

Which of the following is NOT a property of the definite integral?

Accepted Answer

Symmetry

Answer

Linearity

Answer

Additivity

Answer

Anti-derivative

Question 3

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

Accepted Answer

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

Answer

x^3 + e^x + cos(x) + C

Answer

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

Answer

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

Question 4

To calculate the area of the region bounded by the curves y = x² and y = 4, which definite integral should be used?

Accepted Answer

∫[0,2] (4 - x²) dx

Answer

∫[0,2] (x² - 4) dx

Answer

∫[2,0] (4 - x²) dx

Answer

∫[2,0] (x² - 4) dx

Question 5

A force of F(x) = 2x³ + 5x is applied to an object that moves along a straight line from x = 0 to x = 2. To calculate the work done by the force, which integral should be used?

Accepted Answer

∫[0,2] (2x³ + 5x) dx

Answer

∫[2,0] (2x³ + 5x) dx

Answer

∫[0,2] (2x³ - 5x) dx

Answer

∫[2,0] (2x³ - 5x) dx

Question 6

To find the center of mass of a thin rod of length L with density ρ(x) = x, which integral should be used?

Accepted Answer

∫[0,L] xρ(x) dx / ∫[0,L] ρ(x) dx

Answer

∫[0,L] ρ(x) dx / ∫[0,L] xρ(x) dx

Answer

∫[L,0] xρ(x) dx / ∫[L,0] ρ(x) dx

Answer

∫[L,0] ρ(x) dx / ∫[L,0] xρ(x) dx

Question 7

The velocity of a particle is given by v(t) = 3t² - 2t + 1. If the particle starts at position s(0) = 5, which integral should be used to find its position at time t = 3?

Accepted Answer

∫[0,3] (3t² - 2t + 1) dt + 5

Answer

∫[3,0] (3t² - 2t + 1) dt + 5

Answer

∫[0,3] (3t² + 2t + 1) dt + 5

Answer

∫[3,0] (3t² + 2t + 1) dt + 5

Question 8

A force F(x) (in Newtons) acts on a particle as it moves along a straight line. Calculate the work done by the force as the particle moves from position 'a' to position 'b'.

Accepted Answer

∫ F(x) dx from a to b

Answer

∫ F'(x) dx from a to b

Answer

∫ dx / [F(x) from a to b]

Answer

∫ F(a) / [b-a]

Question 9

Find the area under the curve of y = x^2 from x = 0 to x = 2 using definite integral notation.

Accepted Answer

∫[0, 2] x^2 dx

Answer

∫[2, 0] x^2 dx

Answer

2∫[0, 2] x dx

Answer

2∫[2, 0] x dx

Question 10

Calculate the work done in moving an object along the x-axis from x = 0 to x = 3 under the influence of a force F(x) = x^3 + 2x using definite integrals.

Accepted Answer

∫[0, 3] (x^3 + 2x) dx

Answer

∫[3, 0] (x^3 + 2x) dx

Answer

∫[0, 3] x^3 dx + ∫[0, 3] 2x dx

Answer

∫[3, 0] x^3 dx + ∫[3, 0] 2x dx

Question 11

Find the average value of the function f(x) = sin(x) on the interval [0, π/2] using definite integrals.

Accepted Answer

(1/π)∫[0, π/2] sin(x) dx

Answer

π∫[0, π/2] sin(x) dx

Answer

∫[0, π/2] sin(x) dx

Answer

2∫[0, π/2] sin(x) dx

Question 12

Find the average value of the function f(x) = e^x on the interval [0, 1] using definite integrals.

Accepted Answer

(1/e)∫[0, 1] e^x dx

Answer

∫[0, 1] e^x dx

Answer

e∫[0, 1] e^x dx

Answer

2∫[0, 1] e^x dx

Question 13

Determine the volume of the solid generated by rotating the region bounded by the curves y = x^3 and y = 2x about the x-axis from x = 0 to x = 1 using definite integrals.

Accepted Answer

π∫[0, 1] (2x - x^3) dx

Answer

π∫[0, 1] (x^3 - 2x) dx

Answer

2π∫[0, 1] (2x - x^3) dx

Answer

2π∫[0, 1] (x^3 - 2x) dx

Question 14

Calculate the work done by a force of F(x) = cos(x) acting on an object moving along the x-axis from x = 0 to x = π/2 using definite integrals.

Accepted Answer

∫[0, π/2] cos(x) dx

Answer

-∫[0, π/2] cos(x) dx

Answer

-∫[π/2, 0] cos(x) dx

Answer

∫[π/2, 0] cos(x) dx

Question 15

Find the area of the region bounded by the curves y = x^2 and y = 2x + 1 on the interval [0, 3] using the definite integral.

Accepted Answer

∫[0,3] (2x + 1 - x^2) dx

Answer

∫[0,3] (x^2 - 2x - 1) dx

Answer

∫[3,0] (2x + 1 - x^2) dx

Answer

∫[3,0] (x^2 - 2x - 1) dx

Question 16

Determine the work (in joules) done by a force of f(x) = x^3 + 2x on an object moving along the x-axis from x = 0 to x = 2 using the definite integral.

Accepted Answer

∫[0,2] (x^3 + 2x) dx

Answer

∫[2,0] (x^3 + 2x) dx

Answer

∫[0,2] (dx / (x^3 + 2x))

Answer

∫[2,0] (dx / (x^3 + 2x))

Question 17

Determine the center of mass for a thin rod of length 10 with a density function ρ(x) = x.

Accepted Answer

3.33

Answer

2.5

Answer

5

Answer

10

Question 18

Find the volume of the solid obtained by rotating the region bounded by the curves y = cos(x) and y = 0 on the interval [0, π/2] about the x-axis.

Accepted Answer

π

Answer

π/2

Answer

2π

Answer

4π

Question 19

Determine the center of mass of the region bounded by the curves y = x^2 and y = 2x with respect to the y-axis.

Accepted Answer

4/5

Answer

5/4

Answer

3/2

Answer

2/3

Question 20

Find the area of the surface generated by rotating the curve y = x^3 on the interval [0, 1] about the x-axis.

Accepted Answer

π/2

Answer

π/4

Answer

π

Answer

2π

Question 21

Calculate the work done by a force of 100 Newtons acting over a distance of 5 meters in the direction of the force.

Accepted Answer

1000 Joules

Answer

500 Joules

Answer

1500 Joules

Answer

2000 Joules

Question 22

Determine whether the series ∑[n=1, ∞) (1/n) is convergent or divergent.

Accepted Answer

Divergent

Answer

Convergent

Answer

Cannot be determined

Question 23

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

Accepted Answer

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

Answer

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

Answer

x^2 - sin(x) + C

Answer

x^2 + cos(x) + C

Question 24

Which integral represents the area under the curve y = sin(x) between x = 0 and x = π?

Accepted Answer

∫[0, π] sin(x) dx

Answer

∫[0, π/2] sin(x) dx

Answer

∫[π/2, π] sin(x) dx

Answer

∫[π, 0] sin(x) dx

Question 25

A force of F(x) = 100 + 2x newtons is applied to an object moving along a straight line. How much work is done by the force over a distance of 5 meters?

Accepted Answer

∫[0, 5] (100 + 2x) dx

Answer

∫[0, 5] (100 - 2x) dx

Answer

∫[0, 5] 100 dx

Answer

∫[0, 5] 2x dx

Question 26

What is the average value of the function f(x) = x³ on the interval [0, 2]?

Accepted Answer

(1/2) ∫[0, 2] x³ dx

Answer

(0 + 2) / 2

Answer

2∫[0, 2] x³ dx

Answer

∫[0, 2] x³ dx

Question 27

Which of the following is the formula for the Riemann sum of a function f(x) over the interval [a, b] with n subintervals of equal width?

Accepted Answer

∑[i=1, n] f(xᵢ) Δx

Answer

lim[n→∞] ∑[i=1, n] f(xᵢ) Δx

Answer

∫[a, b] f(x) dx

Answer

n∫[a, b] f(x) dx

Question 28

Which of the following is the formula for the area under the curve of a function f(x) over the interval [a, b]?

Accepted Answer

∫[a, b] f(x) dx

Answer

lim[n→∞] ∑[i=1, n] f(xᵢ) Δx

Answer

∑[i=1, n] f(xᵢ) Δx

Answer

n∫[a, b] f(x) dx

Question 29

Which of the following is the formula for the volume of a solid of revolution generated by rotating the region bounded by the curve y = f(x) and the x-axis from x = a to x = b about the x-axis?

Accepted Answer

∫[a, b] π(f(x))² dx

Answer

∫[a, b] 2πf(x) dx

Answer

∫[a, b] π(2f(x))² dx

Question 30

A region is bounded by the curves y = x² and y = 2x. What is the area of this region?

Accepted Answer

∫₀² (2x - x²) dx

Answer

∫₀¹ (2x - x²) dx

Answer

∫₀² (x² + 2x) dx

Answer

∫₀¹ (x² - 2x) dx

Question 31

A cylindrical tank with radius 2 meters and height 5 meters is filled with water. How much work is required to pump all the water out over the top edge of the tank?

Accepted Answer

9800π∫₀⁵ (5-y)y dy

Answer

9800π∫₀⁵ y dy

Answer

9800π∫₀⁵ y(5-y) dy

Answer

9800π∫₀⁵ (5-y) dy

Question 32

The arc length of the curve y = f(x) from x = a to x = b is given by:

Accepted Answer

∫ₐᵇ √(1 + [f'(x)]²) dx

Answer

∫ₐᵇ √(1 + [f(x)]²) dx

Answer

∫ₐᵇ √(1 - [f'(x)]²) dx

Question 33

A rectangular plate with constant density is submerged vertically in water. The plate is 2 meters wide and 3 meters high, with its top edge 1 meter below the surface of the water. What is the hydrostatic force on one side of the plate?

Accepted Answer

9800∫₀³ 2(3-y) dy

Answer

9800∫₀³ 3y dy

Answer

9800∫₀³ 2y dy

Answer

9800∫₀³ 3(3-y) dy