Multiple Integrals: Definition, Applications, and Practice Quizzes

Question 1

Which of the following is a necessary condition for the existence of a double integral over a region R?

Accepted Answer

The function must be continuous on R.

Answer

The region R must be closed and bounded.

Answer

The region R must be simply connected.

Answer

The function must be nonnegative on R.

Question 2

Calculate the surface area of the portion of the paraboloid $z = x^2 + y^2$ that lies inside the cylinder $x^2 + y^2 = 1$.

Accepted Answer

$pi/2$

Answer

$pi$

Answer

2$pi$

Answer

$pi/3$

Question 3

Evaluate the line integral $int_C (x^2 + y) ds$, where $C$ is the curve given by the parametric equations $x = t$ and $y = t^2$ from the point $(0, 0)$ to $(1, 1)$.

Accepted Answer

2/3

Answer

1/3

Answer

1

Answer

2

Question 4

What is the process of evaluating a multiple integral called?

Accepted Answer

Iterated integration

Answer

Summation

Answer

Integration by parts

Answer

Differentiation

Question 5

Which of the following is the correct order of integration for the integral ∫∫R f(x, y) dA, where R is the region bounded by the curves y = x^2 and y = 4?

Accepted Answer

dx dy

Answer

dy dx

Answer

dx dy dx

Answer

dy dx dy

Question 6

Which of the following integrals represents the volume of the solid bounded by the surfaces z = x^2 + y^2 and z = 4?

Accepted Answer

∫∫R (4 - x^2 - y^2) dA

Answer

∫∫R (x^2 + y^2 - 4) dA

Answer

∫∫R (4 - x^2 + y^2) dA

Answer

∫∫R (x^2 - y^2 - 4) dA

Question 7

Which of the following is a fundamental theorem of multiple integrals?

Accepted Answer

The Substitution Rule for Multiple Integrals

Answer

The Fundamental Theorem of Calculus

Answer

The Chain Rule for Multiple Integrals

Answer

The Product Rule for Multiple Integrals

Question 8

What is the main advantage of using iterated integrals to evaluate multiple integrals?

Accepted Answer

It reduces the computation to a sequence of simpler one-dimensional integrals

Answer

It makes it easier to visualize the region of integration

Answer

It provides a more accurate approximation of the integral

Answer

It eliminates the need to find the antiderivative of the function

Question 9

Which of the following statements about double integrals is true?

Accepted Answer

They can be used to find areas of surfaces.

Answer

They can only be used to find volumes.

Answer

They always yield the same result regardless of the order of integration.

Answer

They are only defined for functions of two variables.

Question 10

Which theorem guarantees that a continuous function on a closed interval has an antiderivative?

Accepted Answer

Fundamental Theorem of Calculus

Answer

Chain Rule

Answer

Mean Value Theorem

Answer

Integration by Parts

Question 11

What is the surface area of the surface given by z = f(x, y) over a region R?

Accepted Answer

∫∫[R] √(1 + (∂f/∂x)^2 + (∂f/∂y)^2) dA

Answer

∫∫[R] √(1 - (∂f/∂x)^2 - (∂f/∂y)^2) dA

Answer

∫∫[R] (∂f/∂x)^2 + (∂f/∂y)^2 dA

Answer

∫∫[R] (∂f/∂x) + (∂f/∂y) dA

Question 12

What is the key difference between a double integral and a triple integral?

Accepted Answer

A double integral involves integrating with respect to two variables, while a triple integral involves integrating with respect to three variables.

Answer

A double integral is over a region in the xy-plane, while a triple integral is over a region in 3-dimensional space.

Answer

A double integral is always positive, while a triple integral can be negative.

Answer

A double integral can be evaluated using the Fundamental Theorem of Calculus, while a triple integral cannot.

Question 13

Which of the following applications of multiple integrals is used to calculate the center of mass of a solid?

Accepted Answer

Finding the center of mass of a solid

Answer

Finding the volume of a solid

Answer

Finding the surface area of a surface

Answer

Finding the moment of inertia of a solid

Question 14

Evaluate the line integral $\int_C y dx + x dy$, where $C$ is the circle $x^2 + y^2 = 4$.

Accepted Answer

4π

Answer

0

Answer

2π

Answer

8π

Question 15

Which of the following is a sufficient condition for a function $f(x, y)$ to be differentiable?

Accepted Answer

$f(x, y)$ is continuous and has partial derivatives

Answer

$f(x, y)$ is continuous

Answer

$f(x, y)$ has partial derivatives

Answer

$f(x, y)$ is continuous at every point

Question 16

Set up, but do not evaluate, an integral to find the volume of the solid region inside the cylinder ( x^2 + y^2 = 4 ) and below the paraboloid ( z=x^2+y^2 ) using cylindrical coordinates.

Accepted Answer

$$iint_D int_0^{2pi}int_0^2 r^2 dr d	heta dz$$

Answer

$$iiint_D(x^2+y^2)dxdydz$$

Answer

$$iint_D int_0^{pi/2}int_0^2 r^2 dr d	heta dz$$

Question 17

Evaluate the double integral ∬R f(x, y) dA over the region R bounded by the curves y = x^2 and y = 2x.

Accepted Answer

∫[0, 2] ∫[x^2, 2x] f(x, y) dy dx

Answer

∫[2, 4] ∫[0, x] f(x, y) dy dx

Answer

∫[0, 1] ∫[x^2, 2x] f(x, y) dy dx

Answer

∫[0, 2] ∫[0, y] f(x, y) dy dx

Question 18

Evaluate the surface area of the surface given by z = xy over the rectangular region R = [0, 2] x [0, 3].

Accepted Answer

∫∫R √(1 + x^2 + y^2) dA

Answer

∫∫R (x^2 + y^2) dA

Answer

∫∫R 1 dA

Answer

∫∫R xy dA

Question 19

Find the flux of the vector field F(x, y, z) = (x, y, z) through the surface S, which is the boundary of the solid region E given by 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and 0 ≤ z ≤ 1.

Accepted Answer

∫∫S (x + y + z) dS

Answer

∫∫S z dS

Answer

∫∫S y dS

Answer

∫∫S x dS

Question 20

Evaluate the line integral ∮C (x^2 + y^2) ds where C is the circle of radius 2 centered at the origin.

Accepted Answer

8π

Answer

16π

Answer

32π

Answer

4π

Question 21

Evaluate the line integral ∫C y dx + x dy, where C is the parabolic arc defined by y = x² from the point (0, 0) to (1, 1).

Accepted Answer

2/3

Answer

2

Answer

1/3

Answer

1

Question 22

Is the following statement true or false? The double integral of a function over a region can be evaluated as an iterated integral in any order.

Accepted Answer

True

Answer

False

Answer

It can only be evaluated as an iterated integral in the order dx dy.

Answer

It depends on the function and the region.

Question 23

Find the value of the double integral ∬[0,1][0,x] (y^2 + 2x) dy dx.

Accepted Answer

1/6

Answer

1

Answer

1/2

Answer

1/3

Question 24

Determine if the double integral ∬R (x^3 + y^3) dA converges or diverges, where R is the region bounded by the circle x^2 + y^2 ≤ 1.

Accepted Answer

Converges

Answer

Diverges

Answer

None of the above

Answer

Cannot be determined

Question 25

Find the center of mass of the solid region E bounded by the coordinate planes and the paraboloid z = 4 - x^2 - y^2, with mass density ρ(x, y, z) = z.

Accepted Answer

(0, 0, 8/5)

Answer

(0, 0, 4/5)

Answer

(0, 0, 16/5)

Answer

(0, 0, 32/5)

Question 26

Determine the volume of the solid region bounded by the surfaces z = x^2 + y^2 and z = 4.

Accepted Answer

32π/3

Answer

32π

Answer

16π/3

Answer

16π

Question 27

Convert the iterated integral ∫(∫(∫f(x, y, z) dz) dy) dx into an equivalent triple integral.

Accepted Answer

∭f(x, y, z) dV

Answer

∭f(z, y, x) dV

Answer

∫∫f(x, y, z) dy dz

Answer

∫∫f(z, y, x) dy dz

Question 28

Evaluate the line integral ∫C (x^2 + y^2) ds along the curve C given by the parametric equations x = cos(t) and y = sin(t), where 0 ≤ t ≤ π/2.

Accepted Answer

π

Answer

2π

Answer

π/2

Question 29

Find the flux of the vector field F(x, y, z) = (x, y, z) across the surface S given by z = 4 - x^2 - y^2.

Accepted Answer

-32π

Answer

16π

Answer

0

Question 30

Determine whether the vector field F(x, y, z) = (x, y, z) is conservative or not.

Accepted Answer

Conservative

Answer

Partially Conservative

Answer

Not Conservative

Answer

None of the above

Question 31

Evaluate the double integral ∬R (x + y) dA over the rectangular region R with vertices (0,0), (2,0), (2,1), and (0,1).

Accepted Answer

3

Answer

2

Answer

4

Question 32

Find the volume of the solid bounded by the paraboloid z = x² + y² and the plane z = 4.

Accepted Answer

8π

Answer

32π

Answer

16π

Answer

4π

Question 33

Rewrite the double integral ∫_0^1 ∫_y^1 f(x,y) dx dy with the order of integration reversed.

Accepted Answer

∫_0^1 ∫_0^x f(x,y) dy dx

Answer

∫_0^1 ∫_0^y f(x,y) dx dy

Answer

∫_0^1 ∫_x^1 f(x,y) dy dx

Question 34

Calculate the surface area of the portion of the paraboloid z = 1 - x² - y² that lies above the xy-plane.

Accepted Answer

π(5√5 - 1)/6

Answer

π(5√5 + 1)/3

Answer

π(5√5 - 1)/3

Answer

π(5√5 + 1)/6

Question 35

Which statement is **NOT** a property of double integrals over a region R?

Accepted Answer

If f(x,y) ≤ g(x,y) for all (x,y) in R, then ∫∫_R f(x,y) dA = ∫∫_R g(x,y) dA.

Answer

∫∫_R [f(x,y) + g(x,y)] dA = ∫∫_R f(x,y) dA + ∫∫_R g(x,y) dA

Answer

∫∫_R cf(x,y) dA = c∫∫_R f(x,y) dA, where c is a constant

Question 36

Geometrically, what does a double integral represent?

Accepted Answer

The volume of the solid under a surface and above a region in the xy-plane

Answer

The area of a region in the xy-plane

Answer

The length of a curve in space

Question 37

Determine the correct order of integration for ∫∫_R f(x,y) dA, where R is the region bounded by the curves y = x² and y = 2x.

Accepted Answer

∫_0^2 ∫_x^2^(2x) f(x,y) dy dx

Answer

∫_x^2^(2x) ∫_0^2 f(x,y) dx dy

Answer

∫_0^4 ∫_√y^(y/2) f(x,y) dx dy

Answer

∫_0^2 ∫_0^(2x) f(x,y) dy dx

Question 38

Evaluate the double integral ∫∫_R (x^2 + y^2) dA, where R is the unit circle centered at the origin.

Accepted Answer

π/2

Answer

2π

Answer

π

Answer

0

Question 39

In a change of variables for a double integral, what does the Jacobian determinant represent?

Accepted Answer

The factor by which the area is scaled between the original and transformed regions

Answer

The distance between two points after transformation

Answer

The angle of rotation in the transformation

Question 40

Which of the following is an application of triple integrals?

Accepted Answer

Computing the mass of a solid object with varying density

Answer

Determining the area between two curves

Answer

Finding the length of a curve in space

Question 41

What is the volume of the solid enclosed by the planes z = 0, x = 0, y = 0, and x + y + z = 1?

Accepted Answer

1/6

Answer

1

Answer

1/2

Answer

1/3

Question 42

Which coordinate system is best suited for calculating the triple integral of a function over a sphere?

Accepted Answer

Spherical Coordinates

Answer

Polar Coordinates

Answer

Cartesian Coordinates

Answer

Cylindrical Coordinates

Question 43

What does the integral ∫∫_S f(x,y,z) dS represent, where S is a surface in space?

Accepted Answer

The surface integral of f over S

Answer

The area of the surface S

Answer

The volume enclosed by the surface S

Answer

The line integral of f along the boundary of S

Question 44

What is the relationship between a double integral and a line integral?

Accepted Answer

Green's Theorem relates certain line integrals to double integrals over the enclosed region.

Answer

A double integral is a special case of a line integral.

Answer

They are both used to calculate volumes of solids.

Question 45

What is 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

2

Answer

3

Answer

1