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05 Multiple Integrals Online Quiz Questions

Use this free practice quiz with 20 questions to review 05 Multiple Integrals, test your knowledge, and prepare for your next test or exam.

20 questions
01
Choose one
1 point

Which expression gives the average value of ff over a planar region DD?

  1. A

    ∬Df(x,y) dA\iint_D f(x,y)\,dA

  2. B

    Area⁡(D)∬Df(x,y) dA\operatorname{Area}(D)\iint_D f(x,y)\,dA

  3. C

    1Area⁡(D)∬Df(x,y) dA\frac{1}{\operatorname{Area}(D)}\iint_D f(x,y)\,dA

  4. D

    1∬Df(x,y) dAArea⁡(D)\frac{1}{\iint_D f(x,y)\,dA}\operatorname{Area}(D)

02
Choose one
1 point

For a lamina with mass mm, which formula gives its x-coordinate of the center of mass?

  1. A

    xˉ=Mym\bar{x}=\frac{M_y}{m}

  2. B

    xˉ=Mxm\bar{x}=\frac{M_x}{m}

  3. C

    xˉ=mMy\bar{x}=\frac{m}{M_y}

  4. D

    xˉ=MxMy\bar{x}=\frac{M_x}{M_y}

03
Choose all
1 point

Select all statements that correctly describe mass or average value for a solid region.

  1. A

    For a solid with density ρ(x,y,z)\rho(x,y,z), m=∭Eρ(x,y,z) dVm=\iiint_E \rho(x,y,z)\,dV.

  2. B

    For a solid with constant density ρ0\rho_0, m=ρ0Area⁡(E)m=\rho_0\operatorname{Area}(E).

  3. C

    For constant density ρ0\rho_0, m=ρ0Vol⁡(E)m=\rho_0\operatorname{Vol}(E).

  4. D

    The average value over a solid EE is 1Vol⁡(E)∭Ef dV\frac{1}{\operatorname{Vol}(E)}\iiint_E f\,dV.

04
Choose all
1 point

Select all valid principles for setting up a multiple integral over a nonrectangular region or solid.

  1. A

    For a triple integral, projecting the solid onto a coordinate plane can help determine bounds.

  2. B

    Outer bounds cannot depend on an inner integration variable.

  3. C

    Every multiple integral must use the order dx dy dzdx\,dy\,dz.

  4. D

    For volume between surfaces, the integrand should be top minus bottom.

05
True or false
1 point

True or false: If ff is continuous on a rectangular region, reversing the order of its double integral leaves the value unchanged.

  1. A

    True

  2. B

    False

06
True or false
1 point

True or false: If f(x,y)≥g(x,y)f(x,y)\ge g(x,y) on a planar region DD, then the volume between z=f(x,y)z=f(x,y) and z=g(x,y)z=g(x,y) is ∬D[f(x,y)−g(x,y)] dA\iint_D [f(x,y)-g(x,y)]\,dA.

  1. A

    True

  2. B

    False

07
Written response
1 point

Evaluate the rectangular-region integral ∫01∫02(3x+y) dy dx\int_0^1\int_0^2 (3x+y)\,dy\,dx. Enter the exact numerical value.

08
Written response
1 point

What is the standard term for the function ρ(x,y)\rho(x,y) that describes how mass is distributed across a thin plate?

09
Fill in the blank
1 point

Complete the general setup for a solid E={(x,y,z):(x,y)∈D, u(x,y)≤z≤v(x,y)}E=\{(x,y,z):(x,y)\in D,\,u(x,y)\le z\le v(x,y)\}: \iiint_E f(x,y,z)\,dV=\iint_D\int_^f(x,y,z) dz dAf(x,y,z)\,dz\,dA. Enter the lower and upper z-bound expressions.

10
Fill in the blank
1 point

Complete the center-of-mass formulas for a lamina: xˉ=\bar{x}= and yˉ=\bar{y}=.

11
Open ended
1 point

Set up, but do not evaluate, a triple integral for the volume of the solid in the first octant below the plane z=4−2x−yz=4-2x-y and above the triangular projection x≥0x\ge0, y≥0y\ge0, x+y≤2x+y\le2. Explain how each set of bounds is obtained.

12
Choose one
1 point

What is the value of ∭B1 dV\iiint_B 1\,dV for the box B=[−1,2]×[1,3]×[2,5]B=[-1,2]\times[1,3]\times[2,5]?

  1. A

    66

  2. B

    1818

  3. C

    2424

  4. D

    3030

13
Choose one
1 point

Which integral gives the area of a planar region DD?

  1. A

    ∬Df(x,y) dA\iint_D f(x,y)\,dA

  2. B

    ∬D1 dA\iint_D 1\,dA

  3. C

    ∭E1 dV\iiint_E 1\,dV

  4. D

    ∬Dρ(x,y) dA\iint_D \rho(x,y)\,dA

14
True or false
1 point

True or false: If ff is continuous on a rectangular region, then reversing the order of integration in its double integral leaves the value unchanged.

  1. A

    True

  2. B

    False

15
Choose one
1 point

Which iterated integral correctly represents ∬Df(x,y) dA\iint_D f(x,y)\,dA for D={(x,y):x≥0, y≥0, x+y≤4}D=\{(x,y):x\ge0,\ y\ge0,\ x+y\le4\}, using the order dy dxdy\,dx?

  1. A

    ∫04∫0xf(x,y) dy dx\int_0^4\int_0^x f(x,y)\,dy\,dx

  2. B

    ∫04∫x4f(x,y) dy dx\int_0^4\int_x^4 f(x,y)\,dy\,dx

  3. C

    ∫04∫04−xf(x,y) dy dx\int_0^4\int_0^{4-x} f(x,y)\,dy\,dx

  4. D

    ∫04∫04+xf(x,y) dy dx\int_0^4\int_0^{4+x} f(x,y)\,dy\,dx

16
Written response
1 point

Find the volume of the rectangular box B=[0,2]×[1,3]×[−1,1]B=[0,2]\times[1,3]\times[-1,1]. Enter the numerical value.

17
Choose one
1 point

A lamina occupies a planar region DD with constant surface density ρ0\rho_0. Which expression gives its mass?

  1. A

    m=ρ0+Area⁡(D)m=\rho_0+\operatorname{Area}(D)

  2. B

    m=Area⁡(D)ρ0m=\frac{\operatorname{Area}(D)}{\rho_0}

  3. C

    m=ρ0Area⁡(D)m=\rho_0\operatorname{Area}(D)

  4. D

    m=ρ0Vol⁡(D)m=\rho_0\operatorname{Vol}(D)

18
Written response
1 point

In the formula for the average value of ff over a solid region EE, what geometric quantity appears in the denominator? Enter the quantity as one word.

19
Choose one
1 point

What is the volume of the solid in the first octant below z=6−2x−3yz=6-2x-3y?

  1. A

    22

  2. B

    33

  3. C

    44

  4. D

    66

20
Choose one
1 point

For a lamina with total mass mm, which formula correctly gives its xx-coordinate of the center of mass?

  1. A

    xˉ=Mym\bar{x}=\frac{M_y}{m}

  2. B

    xˉ=Mxm\bar{x}=\frac{M_x}{m}

  3. C

    yˉ=Mym\bar{y}=\frac{M_y}{m}

  4. D

    xˉ=mMy\bar{x}=mM_y