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06 Multiple Integrals in Curvilinear Coordinates Online Quiz Questions

Use this free practice quiz with 20 questions to review 06 Multiple Integrals in Curvilinear Coordinates, test your knowledge, and prepare for your next test or exam.

20 questions
01
True or false
1 point

True or false: In polar coordinates, the area element is dA=r dr dθdA=r\,dr\,d\theta, rather than simply dr dθdr\,d\theta.

  1. A

    True

  2. B

    False

02
Choose one
1 point

Which set of equations gives the standard spherical-coordinate conversion used in the material?

  1. A

    x=ρcos⁡ϕx=\rho\cos\phi, y=ρsin⁡ϕy=\rho\sin\phi, z=ρz=\rho

  2. B

    x=ρsin⁡θcos⁡ϕx=\rho\sin\theta\cos\phi, y=ρsin⁡θsin⁡ϕy=\rho\sin\theta\sin\phi, z=ρcos⁡θz=\rho\cos\theta

  3. C

    x=ρsin⁡ϕcos⁡θx=\rho\sin\phi\cos\theta, y=ρsin⁡ϕsin⁡θy=\rho\sin\phi\sin\theta, z=ρcos⁡ϕz=\rho\cos\phi

  4. D

    x=ρcos⁡θx=\rho\cos\theta, y=ρsin⁡θy=\rho\sin\theta, z=ρsin⁡ϕz=\rho\sin\phi

03
Fill in the blank
1 point

Complete the formula for the polar-coordinate area element: dA=dA= dr dθ\,dr\,d\theta.

04
Written response
1 point

What factor multiplies dρ dϕ dθd\rho\,d\phi\,d\theta in the spherical-coordinate volume element?

05
Choose one
1 point

Which cylindrical-coordinate bounds describe the upper hemisphere x2+y2+z2≤R2x^2+y^2+z^2\le R^2, z≥0z\ge0, when the integration order is dz dr dθdz\,dr\,d\theta?

  1. A

    In the order dz dr dθdz\,dr\,d\theta: 0≤z≤R2−r20\le z\le\sqrt{R^2-r^2}, 0≤r≤R0\le r\le R, 0≤θ≤π0\le\theta\le\pi.

  2. B

    In the order dz dr dθdz\,dr\,d\theta: 0≤z≤R2−r20\le z\le\sqrt{R^2-r^2}, 0≤r≤R0\le r\le R, 0≤θ≤2π0\le\theta\le2\pi.

  3. C

    In the order dz dr dθdz\,dr\,d\theta: −R2−r2≤z≤R2−r2-\sqrt{R^2-r^2}\le z\le\sqrt{R^2-r^2}, 0≤r≤R0\le r\le R, 0≤θ≤2π0\le\theta\le2\pi.

  4. D

    In the order dz dr dθdz\,dr\,d\theta: 0≤z≤R−r0\le z\le R-r, 0≤r≤R20\le r\le R^2, 0≤θ≤2π0\le\theta\le2\pi.

06
True or false
1 point

True or false: In the standard spherical convention, the cone z=x2+y2z=\sqrt{x^2+y^2} corresponds to ϕ=π/4\phi=\pi/4, and the region above the cone has 0≤ϕ≤π/40\le\phi\le\pi/4.

  1. A

    True

  2. B

    False

07
Choose all
1 point

Select all correct statements about the coordinate systems described in the material.

  1. A

    Polar coordinates use x=rcos⁡θx=r\cos\theta and y=rsin⁡θy=r\sin\theta.

  2. B

    Spherical coordinates always use z=ρsin⁡ϕz=\rho\sin\phi.

  3. C

    Cylindrical coordinates have volume element r dr dθ dzr\,dr\,d\theta\,dz.

  4. D

    Spherical coordinates have volume element ρ2sin⁡ϕ dρ dϕ dθ\rho^2\sin\phi\,d\rho\,d\phi\,d\theta.

08
Fill in the blank
1 point

For a constant-density solid, completing the cylindrical-coordinate expression for the moment of inertia about the zz-axis gives a total radial factor of in the integrand.

09
Written response
1 point

For the planar region x2+y2≤2axx^2+y^2\le2ax, what is the polar-coordinate upper bound for rr?

10
Choose one
1 point

Which iterated integral correctly represents ∬D(x2+y2) dA\iint_D(x^2+y^2)\,dA over the disk D:x2+y2≤4D:x^2+y^2\le4 in polar coordinates?

  1. A

    ∫02π∫02r3 dr dθ\int_0^{2\pi}\int_0^2 r^3\,dr\,d\theta

  2. B

    ∫02π∫02r2 dr dθ\int_0^{2\pi}\int_0^2 r^2\,dr\,d\theta

  3. C

    ∫04∫02πr3 dθ dr\int_0^4\int_0^{2\pi} r^3\,d\theta\,dr

  4. D

    ∫02π∫02r4 dr dθ\int_0^{2\pi}\int_0^2 r^4\,dr\,d\theta

11
Choose all
1 point

Select all steps that are required when setting up a multiple integral after changing coordinates.

  1. A

    Rewrite the integrand in the new coordinates.

  2. B

    Transform the integrand but leave the original differential element unchanged.

  3. C

    Determine bounds that cover the intended region exactly once.

  4. D

    Insert the appropriate absolute Jacobian factor.

12
Open ended
1 point

Set up, and if possible evaluate, the volume of the region inside the sphere x2+y2+z2≤R2x^2+y^2+z^2\le R^2 and above the cone z=x2+y2z=\sqrt{x^2+y^2}. Use the standard spherical-coordinate convention from the material.

13
Choose one
1 point

What is the volume of the upper hemisphere x2+y2+z2≤R2x^2+y^2+z^2\le R^2, z≥0z\ge0, when evaluated using cylindrical coordinates?

  1. A

    πR3\pi R^3

  2. B

    23πR3\frac{2}{3}\pi R^3

  3. C

    43πR3\frac{4}{3}\pi R^3

  4. D

    2πR32\pi R^3

14
Choose one
1 point

When converting a planar integral from rectangular coordinates to polar coordinates, which differential area element correctly accounts for local area scaling?

  1. A

    dA=dr dθdA=dr\,d\theta

  2. B

    dA=r dr dθdA=r\,dr\,d\theta

  3. C

    dA=r2 dr dθdA=r^2\,dr\,d\theta

  4. D

    dA=sin⁡θ dr dθdA=\sin\theta\,dr\,d\theta

15
Choose one
1 point

Which polar-coordinate bounds describe the disk x2+y2≤2ayx^2+y^2\le 2ay exactly once, assuming a>0a>0?

  1. A

    0≤r≤2acos⁡θ0\le r\le 2a\cos\theta, 0≤θ≤π0\le\theta\le\pi

  2. B

    0≤r≤2asin⁡θ0\le r\le 2a\sin\theta, −π2≤θ≤π2-\frac{\pi}{2}\le\theta\le\frac{\pi}{2}

  3. C

    0≤r≤2asin⁡θ0\le r\le 2a\sin\theta, 0≤θ≤π0\le\theta\le\pi

  4. D

    0≤r≤a0\le r\le a, 0≤θ≤2π0\le\theta\le 2\pi

16
Choose one
1 point

Which cylindrical-coordinate bounds represent the upper hemisphere x2+y2+z2≤R2x^2+y^2+z^2\le R^2, z≥0z\ge0, when the integration order is dθ dz drd\theta\,dz\,dr?

  1. A

    For the order dθ dz drd\theta\,dz\,dr: 0≤r≤R0\le r\le R, 0≤z≤R2−r20\le z\le\sqrt{R^2-r^2}, 0≤θ≤π0\le\theta\le\pi.

  2. B

    For the order dθ dz drd\theta\,dz\,dr: 0≤r≤R0\le r\le R, −R2−r2≤z≤R2−r2-\sqrt{R^2-r^2}\le z\le\sqrt{R^2-r^2}, 0≤θ≤2π0\le\theta\le2\pi.

  3. C

    For the order dθ dz drd\theta\,dz\,dr: 0≤r≤R0\le r\le R, 0≤z≤R2−r20\le z\le\sqrt{R^2-r^2}, 0≤θ≤2π0\le\theta\le2\pi.

  4. D

    For the order dθ dz drd\theta\,dz\,dr: 0≤r≤R20\le r\le R^2, 0≤z≤R−r0\le z\le R-r, 0≤θ≤2π0\le\theta\le2\pi.

17
Choose one
1 point

Using the standard spherical convention in which ϕ\phi is measured downward from the positive zz-axis, which range of ϕ\phi describes the part inside ρ≤R\rho\le R and above the cone z=x2+y2z=\sqrt{x^2+y^2}?

  1. A

    π4≤ϕ≤π\frac{\pi}{4}\le\phi\le\pi

  2. B

    0≤ϕ≤π20\le\phi\le\frac{\pi}{2}

  3. C

    0≤ϕ≤π30\le\phi\le\frac{\pi}{3}

  4. D

    0≤ϕ≤π40\le\phi\le\frac{\pi}{4}

18
True or false
1 point

True or false: If a solid is symmetric about the xyxy-plane and its density is also symmetric about that plane, then its center-of-mass coordinate satisfies zˉ=0\bar z=0.

  1. A

    True

  2. B

    False

19
Written response
1 point

In cylindrical coordinates, the absolute Jacobian is rr. What is its value at the point where r=5r=5?

20
Written response
1 point

In the two-dimensional change-of-variables formula, what local geometric quantity is represented by ∣∂(x,y)∂(u,v)∣\left|\frac{\partial(x,y)}{\partial(u,v)}\right|?