Master Integration Applications: Enhance Your Calculus III Skills

Question 1

Which of the following is NOT an application of integration in Calculus III?

Accepted Answer

Solving differential equations using separation of variables

Answer

Calculating volumes of solids of revolution

Answer

Determining the area of surfaces

Answer

Finding the work done by a force over a distance

Question 2

To find the volume of a solid generated by revolving the region bounded by y = x^2 and y = 4 about the x-axis, which integral should be used?

Accepted Answer

∫[0, 2] π(4 - x^2) dx

Answer

∫[0, 2] 2πx(4 - x^2) dx

Answer

∫[0, 4] π(x^2 - 4) dx

Answer

∫[0, 4] 2πx(x^2 - 4) dx

Question 3

Which of the following is NOT an application of integration in Calculus III?

Accepted Answer

Solving differential equations

Answer

Calculating areas of surfaces

Answer

Finding volumes of solids

Answer

Calculating surface integrals

Question 4

To calculate the area of a surface given by the function f(x, y), which integral should be used?

Accepted Answer

Double integral over the projection of the surface onto the xy-plane

Answer

Single integral with respect to x

Answer

Single integral with respect to y

Answer

Triple integral over the surface

Question 5

What is the physical interpretation of the integral ∫[C] F(r) · dr, where F is a force field and C is a path?

Accepted Answer

The work done by the force field in moving a particle along the path

Answer

The potential energy gained by a particle moving along the path

Answer

The circulation of the force field around the path

Answer

The flux of the force field through the surface bounded by the path

Question 6

Find the work done by the force field F(x, y) = (x^2, y^2) in moving a particle along the curve C given by y = x^3 from (0, 0) to (1, 1).

Accepted Answer

1/6 unit of work

Answer

1/2 unit of work

Answer

1 unit of work

Answer

3/2 units of work

Question 7

Find the surface area of the portion of the paraboloid z = x^2 + y^2 that lies within the cylinder x^2 + y^2 ≤ 4.

Accepted Answer

16π square units

Answer

8π square units

Answer

32π square units

Answer

64π square units

Question 8

Find the work done by a force of f(x) = x^2 + 1 acting on an object that moves along the x-axis from x = 0 to x = 2.

Accepted Answer

∫[0,2] (x^2 + 1) dx

Answer

∫[0,2] x(x^2 + 1) dx

Answer

∫[0,2] (x^2 + 1)^2 dx

Answer

∫[0,2] x^2(x^2 + 1) dx

Question 9

Calculate the circulation of the vector field F = (x^2, y^2) around the circle of radius r centered at the origin.

Accepted Answer

4πr^3

Answer

2πr^3

Answer

πr^3

Answer

8πr^3

Question 10

Determine the potential function f(x, y) for the vector field F(x, y) = (2x + 3y, 4x - y), if it exists.

Accepted Answer

f(x, y) = x^2 + 3xy - y^2

Answer

f(x, y) = 2x^2 + 4xy

Answer

f(x, y) = sin(x) + cos(y)

Answer

f(x, y) cannot be found

Question 11

Precise: Find the surface area generated by rotating the curve y = x³, from x = 0 to x = 1, about the x-axis.

Accepted Answer

π(2√2 - 1)

Answer

2π(2√2 - 1)

Answer

π(4√2 - 1)

Answer

2π(4√2 - 1)

Question 12

Which line integral represents the work done by a constant force F in moving an object along a curve C?

Accepted Answer

∫_C F(r) ⋅ dr

Answer

∫_C |F(r)| dr

Answer

∫_C F(r) × dr

Answer

∫_C F(r) ||dr||

Question 13

Which surface integral represents the flux of a velocity field F across a surface S?

Accepted Answer

∫_S F(r) ⋅ dS

Answer

∫_S F(r) ||dS||

Answer

∫_S |F(r)| dS

Answer

∫_S F(r) × dS

Question 14

Which integral represents the moment of inertia of a region D about the y-axis?

Accepted Answer

∫_D x² dσ

Answer

∫_D x dσ

Answer

∫_D xy dσ

Answer

∫_D y² dσ

Question 15

What is the divergence of a vector field F(x, y, z)?

Accepted Answer

∇ ⋅ F = ∂F_x/∂x + ∂F_y/∂y + ∂F_z/∂z

Answer

∇ ⋅ F = ∂F_x/∂x - ∂F_y/∂y + ∂F_z/∂z

Answer

∇ ⋅ F = ∂F_x/∂z + ∂F_y/∂x + ∂F_z/∂y

Answer

∇ ⋅ F = ∂F_x/∂y + ∂F_y/∂z + ∂F_z/∂x

Question 16

What is the curl of a vector field F(x, y, z)?

Accepted Answer

∇ × F = (∂F_z/∂y - ∂F_y/∂z)i + (∂F_x/∂z - ∂F_z/∂x)j + (∂F_y/∂x - ∂F_x/∂y)k

Answer

∇ × F = (∂F_z/∂y + ∂F_y/∂z)i + (∂F_x/∂z + ∂F_z/∂x)j + (∂F_y/∂x + ∂F_x/∂y)k

Answer

∇ × F = (∂F_z/∂y - ∂F_y/∂z)j + (∂F_x/∂z - ∂F_z/∂x)k + (∂F_y/∂x - ∂F_x/∂y)i

Question 17

Calculate the center of mass of a thin plate with density δ(x, y) = 2xy grams per square meter, where the plate occupies the region bounded by the parabola y = x^2 and the line y = 4.

Accepted Answer

(2/3, 8/5)

Answer

(2/3, 5/8)

Answer

(3/2, 8/5)

Answer

(3/2, 5/8)

Question 18

A cylindrical tank is being filled with water at a constant rate. The base of the tank has a radius of 2 meters and its height is 5 meters. At the moment when the depth of the water is 2 meters, the water level is rising at a rate of 0.5 meters per minute. At this instant, find the rate at which the volume of the water is increasing.

Accepted Answer

14π cubic meters per minute

Answer

16π cubic meters per minute

Answer

20π cubic meters per minute

Answer

8π cubic meters per minute

Question 19

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

Accepted Answer

2/π

Answer

0

Answer

π/2

Answer

1

Question 20

A population of bacteria grows at a rate proportional to its size. If there are initially 1000 bacteria and the population doubles in size every 5 hours, how many bacteria will there be after 10 hours?

Accepted Answer

4000

Answer

16000

Answer

8000

Answer

2000

Question 21

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

Accepted Answer

16π/5

Answer

8π/5

Answer

64π/5

Answer

32π/5

Question 22

Find the surface area of the surface generated by rotating the curve y = x^3 from x = 0 to x = 2 about the x-axis.

Accepted Answer

16π/5

Answer

32π/5

Answer

64π/5

Answer

8π/5

Question 23

A particle moves along a straight line with an acceleration of a(t) = t - 2. If the particle's initial velocity is 5 m/s, find its position function s(t).

Accepted Answer

s(t) = t^2/2 - 2t + 5

Answer

s(t) = -t^2/2 + 2t + 5

Answer

s(t) = t^2/2 + 2t + 5

Answer

s(t) = -t^2/2 - 2t + 5

Question 24

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

Accepted Answer

π

Answer

4π

Answer

2π

Answer

π/2

Question 25

Find the work done in pumping all the water out of a spherical tank with radius 2 meters, given that the density of water is 1000 kg/m^3 and the acceleration due to gravity is 9.8 m/s^2.

Accepted Answer

128000π J

Answer

512000π J

Answer

64000π J

Answer

256000π J

Question 26

A rectangular box with a square base is to be constructed to hold a volume of 1000 cubic centimeters. The material for the base costs $0.10 per square centimeter, and the material for the sides costs $0.05 per square centimeter. Find the dimensions of the box that will minimize the cost of construction.

Accepted Answer

x = y = z = 10 cm

Answer

x = y = 10 cm, z = 5 cm

Answer

x = y = z = 5 cm

Answer

x = y = 5 cm, z = 10 cm

Question 27

Find the area of the surface generated by rotating the curve y = ln(x) from x = 1 to x = 2 about the y-axis.

Accepted Answer

2π(ln(2) + 1)

Answer

2π(ln(2) - 1)

Answer

π(ln(2) + 1)

Answer

π(ln(2) - 1)