Master Multiple Integrals in Engineering Mathematics: A Comprehensive Guide

Question 1

Which technique is not used for evaluating triple integrals?

Accepted Answer

Integration by parts

Answer

Iterated integration

Answer

Change of variables

Answer

Cylindrical coordinates

Question 2

Which application is not related to multiple integrals in engineering?

Accepted Answer

Solving linear equations

Answer

Calculating the volume of a fluid

Answer

Calculating the surface area of a curved surface

Answer

Finding the work done by a force

Question 3

What is the correct formula for changing variables in a double integral?

Accepted Answer

∫∫f(x, y) dx dy = ∫∫f(u, v) |J| du dv

Answer

∫∫f(x, y) dx dy = ∫∫f(u, v) |u - v| du dv

Answer

∫∫f(x, y) dx dy = ∫∫f(u, v) |x - y| du dv

Answer

∫∫f(x, y) dx dy = ∫∫f(u, v) |J|^2 du dv

Question 4

Which property does not apply to double integrals?

Accepted Answer

Symmetry

Answer

Linearity

Answer

Separability

Answer

Additivity

Question 5

Which formula represents the triple integral of a function f(x, y, z) over a solid region E?

Accepted Answer

∫∫∫f(x, y, z) dV = ∫∫∫f(x, y, z) dx dy dz

Answer

∫∫∫f(x, y, z) dV = ∫∫∫f(x, y, z) dy dx dz

Answer

∫∫∫f(x, y, z) dV = ∫∫∫f(x, y, z) dz dy dx

Answer

∫∫∫f(x, y, z) dV = ∫∫∫f(x, y, z) dx dz dy

Question 6

Which of the following change of variables is used to evaluate a double integral over a polar region?

Accepted Answer

Polar coordinates

Answer

Cylindrical coordinates

Answer

Spherical coordinates

Question 7

The line integral of a vector field around a closed curve is always zero if the vector field is conservative.

Accepted Answer

True

Answer

False

Question 8

Evaluate the double integral ∬R (x^2 + y^2) dA, where R is the circular region bounded by the circle x^2 + y^2 = 1.

Accepted Answer

π

Answer

π/2

Answer

2π

Answer

4π

Question 9

Find the surface area of the surface defined by z = xy over the rectangle R: [0, 2] x [0, 1].

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 10

Which of the following is the correct change of variables to evaluate the integral ∫∫[R] e^(x^2+y^2) dA?

Accepted Answer

x = r cos(θ), y = r sin(θ)

Answer

u = x/y, v = y/x

Answer

x = r sin(θ), y = r cos(θ)

Answer

u = x^2+y^2, v = xy

Question 11

Find the triple integral of the function $f(x, y, z) = x$ over the solid region bounded by the surfaces $x^2 + y^2 = 1$, $z = 0$, and $z = x + y$.

Accepted Answer

∫∫∫[D] x dz dy dx

Answer

∫∫∫[D] (x + y) dz dy dx

Answer

∫∫∫[D] x^2 dz dy dx

Answer

∫∫∫[D] xy dz dy dx

Question 12

Which of the following describes the region of integration for the triple integral ∫∫∫[R] f(x, y, z) dV, where R is the solid bounded by the surfaces $z = 0$, $z = 1$, $y = 0$, $y = x$, and $x = 1$?

Accepted Answer

{(x, y, z) | 0 ≤ x ≤ 1, 0 ≤ y ≤ x, 0 ≤ z ≤ 1}

Answer

{(x, y, z) | 0 ≤ x ≤ 1, 0 ≤ z ≤ y, 0 ≤ y ≤ 1}

Answer

{(x, y, z) | 0 ≤ y ≤ 1, 0 ≤ z ≤ x, 0 ≤ x ≤ 1}

Answer

{(x, y, z) | 0 ≤ y ≤ 1, 0 ≤ x ≤ y, 0 ≤ z ≤ 1}

Question 13

Which of the following integrals represents the work done by a force field $F = (2x, -y)$ in moving a particle along the curve $C = x^2 + y^2 = 1$?

Accepted Answer

∫[C] 2x dx - y dy

Answer

∫∫[D] 2x dx dy

Answer

∫[C] x^2 dx + y^2 dy

Question 14

Find the flux of the vector field $F = (x, y)$ across the boundary of the region bounded by the curves $x = 0$, $x = 2$, $y = 0$, and $y = 1$.

Accepted Answer

4

Answer

2

Answer

8

Answer

16

Question 15

Evaluate the double integral ∫∫_R (x² + y²) dA, where R is the region bounded by the circle x² + y² = 4.

Accepted Answer

8π

Answer

16π

Answer

4π

Question 16

Determine the volume of the solid enclosed by the planes z = 0, x = 0, y = 0, and x + y + z = 1.

Accepted Answer

∫₀¹∫₀^1-x(1 - x - y) dy dx

Answer

∫₀¹∫₀^x(1 - x - y) dy dx

Answer

∫₀¹∫₀¹(1 - x - y) dy dx

Question 17

Calculate the Jacobian determinant of the transformation given by u = x + y, v = x - y.

Accepted Answer

-2

Answer

-1

Answer

1

Answer

2

Question 18

What does the double integral ∫∫_S dS represent, where S is a surface in three-dimensional space?

Accepted Answer

Surface area of S

Answer

None of the above

Answer

Perimeter of S

Answer

Volume enclosed by S

Question 19

Evaluate the iterated integral ∫₀²∫₀¹ xy² dx dy.

Accepted Answer

1/3

Answer

1/2

Answer

1

Answer

2/3

Question 20

Set up a double integral representing the area of the region enclosed by the curves y = x² and y = 2x.

Accepted Answer

∫₀² (2x - x²) dx

Answer

∫₀⁴ (x² - 2x) dx

Answer

∫₀² (x² - 2x) dx

Answer

∫₀⁴ (2x - x²) dx

Question 21

Calculate the average value of the function f(x,y) = x + y over the rectangle R = {(x,y) | 0 ≤ x ≤ 2, 0 ≤ y ≤ 1}.

Accepted Answer

3/2

Answer

1

Answer

3

Answer

2

Question 22

Transform the double integral ∫₀¹∫₀^x dy dx into polar coordinates.

Accepted Answer

∫₀^π/4∫₀^{sec θ} r dr dθ

Answer

∫₀^π/4∫₀^{csc θ} r dr dθ

Answer

∫₀^π/2∫₀^{sec θ} r dr dθ

Question 23

Determine the volume of the solid bounded by the paraboloid z = 4 - x² - y² and the xy-plane.

Accepted Answer

8π

Answer

4π

Answer

2π

Answer

16π

Question 24

The integral ∫∫∫_E (x² + y² + z²) dV, where E is the unit sphere, represents:

Accepted Answer

Moment of inertia of E about the origin

Answer

Surface area of E

Answer

Volume of E

Answer

Mass of E