Multiple Integrals: Definition, Iterated Integrals, and Applications

Question 1

Evaluate the following surface integral: ∫∫S (x + y) dS, where S is the part of the plane z = 2x + y that lies above the rectangle R = [0, 1] × [0, 2].

Accepted Answer

10

Answer

6

Answer

8

Answer

12

Question 2

Find the Jacobian of the transformation x = u + v, y = u - v.

Accepted Answer

2

Answer

1

Answer

-1

Answer

-2

Question 3

What is the theorem that allows us to switch the order of integration in a double integral?

Accepted Answer

Fubini's Theorem

Answer

Cauchy's Theorem

Answer

Green's Theorem

Answer

Stokes' Theorem

Question 4

What formula calculates the surface area of the surface given by the function z = f(x, y) over the region R in the xy-plane?

Accepted Answer

∫∫R √(1 + (∂f/∂x)^2 + (∂f/∂y)^2) dA

Answer

∫∫R f(x, y) dA

Answer

∫∫R f(x, y) dx dy

Answer

∫∫R √(1 + (∂f/∂x)^2 + (∂f/∂y)^2) dx dy

Question 5

Which formula represents the flux of the vector field F = (P, Q) across a curve C?

Accepted Answer

∫C P dx + Q dy

Answer

-∫C P dx + Q dy

Answer

∫C P dy - Q dx

Answer

-∫C P dy - Q dx

Question 6

Which formula yields the divergence of the vector field F = (P, Q, R)?

Accepted Answer

∂P/∂x + ∂Q/∂y + ∂R/∂z

Answer

∂P/∂y + ∂Q/∂z + ∂R/∂x

Answer

∂P/∂z + ∂Q/∂x + ∂R/∂y

Answer

∂P/∂x - ∂Q/∂y - ∂R/∂z

Question 7

Establish whether the following vector field is conservative: F = (y, -x).

Accepted Answer

Yes

Answer

No

Question 8

Which of the following is the correct formula for the flux integral ∫∫_S F · n dS, where S is a surface, F is a vector field, and n is the unit normal vector?

Accepted Answer

∫∫_S (F · n) dS

Answer

∫∫_S F · (∇ × F) dS

Answer

∫∫_S n · (∇ × F) dS

Answer

∫∫_S (∇ · F) dS

Question 9

Which of the following is a possible solution to the differential equation y' = xy?

Accepted Answer

y = e^(x^2/2)

Answer

y = x

Answer

y = x^2

Answer

y = 1/x

Question 10

When evaluating multiple integrals of nonnegative functions, what is a key characteristic of their resulting values?

Accepted Answer

They are always nonnegative.

Answer

They can be either positive or negative.

Answer

Their signs depend on the integrand's sign.

Question 11

Evaluate the triple integral ∫∫∫[D (x^2 + y^2)^3 dV, where D is the solid region bounded by the surfaces z = 0, z = 1, and x^2 + y^2 ≤ 4.

Accepted Answer

512π/15

Answer

2048π/5

Answer

1024π/5

Answer

512π/5

Question 12

Calculate the double integral ∬[x^2 + y^2] dA over the region R bounded by the circle x^2 + y^2 = 9.

Accepted Answer

9π

Answer

π(3)/2

Answer

π(3)^2

Answer

2π(3)

Question 13

Evaluate the double integral $\int_{0}^{1} \int_{0}^{x} (x+y) \, dy \, dx$.

Accepted Answer

1

Answer

1/2

Answer

3/4

Answer

2/3

Question 14

Find the volume of the solid bounded by the cylinder $x^2 + y^2 = 4$, the plane $z = 0$, and the plane $z = x + 2$.

Accepted Answer

16π

Answer

32π

Answer

4π

Answer

8π

Question 15

Which of the following integrals represents the surface area of the part of the paraboloid $z = x^2 + y^2$ that lies below the plane $z = 4$?

Accepted Answer

$\int_{0}^{2} \int_{0}^{2π} \sqrt{1 + 4r^2} \, r \, dθ \, dr$

Answer

$\int_{0}^{4} \int_{0}^{2π} \sqrt{1 + 4r^2} \, r \, dθ \, dr$

Answer

$\int_{0}^{2} \int_{0}^{2π} \sqrt{1 + 2r^2} \, r \, dθ \, dr$

Question 16

Change the order of integration for the integral $\int_{0}^{1} \int_{x^2}^{1} f(x,y) \, dy \, dx$.

Accepted Answer

$\int_{0}^{1} \int_{0}^{\sqrt{y}} f(x,y) \, dx \, dy$

Answer

$\int_{0}^{1} \int_{\sqrt{y}}^{1} f(x,y) \, dx \, dy$

Answer

$\int_{0}^{1} \int_{0}^{y^2} f(x,y) \, dx \, dy$

Question 17

Evaluate the triple integral $\int_{0}^{1} \int_{0}^{x} \int_{0}^{y} (x+y+z) \, dz \, dy \, dx$.

Accepted Answer

1/8

Answer

1

Answer

1/2

Answer

1/4

Question 18

Find the volume of the solid bounded by the paraboloid $z = x^2 + y^2$ and the plane $z = 1$.

Accepted Answer

π/2

Answer

2π

Answer

π

Question 19

Which of the following is the correct formula for the Jacobian determinant of the transformation $x = u^2 - v^2$, $y = 2uv$?

Accepted Answer

4(u^2 + v^2)

Answer

2(u^2 + v^2)

Answer

4uv

Question 20

Evaluate the integral $\int_{0}^{1} \int_{0}^{1-x} \int_{0}^{1-x-y} \, dz \, dy \, dx$ using a suitable change of variables.

Accepted Answer

1/6

Answer

1/2

Answer

1/3

Answer

1

Question 21

Find the average value of the function $f(x,y) = x^2 + y^2$ over the region $R$ bounded by the circle $x^2 + y^2 = 1$.

Accepted Answer

1/2

Answer

1/4

Answer

1

Answer

2

Question 22

Which of the following statements is true about the integral $\int_{0}^{1} \int_{0}^{x} \int_{0}^{y} f(x,y,z) \, dz \, dy \, dx$?

Accepted Answer

The order of integration can be changed to $\int_{0}^{1} \int_{z}^{1} \int_{z}^{y} f(x,y,z) \, dx \, dy \, dz$.

Answer

The order of integration cannot be changed.