Triple Integrals and Coordinates for Solids · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Integrating solids and choosing coordinates

Mathematics · Calculus & Analysis · ages 20-21
Name ______________________   Date ____________

  1. What is the volume of the box with 0 <= x <= 2, 0 <= y <= 1, 0 <= z <= 3?

    Answer: ______________

  2. A spherical iterated integral has rho inside, phi in the middle and theta outside. Which differential order matches it?

    • d theta d phi d rho
    • d rho d phi d theta
    • d phi d rho d theta
    • d rho d theta d phi

  3. Find the volume of 0 <= x <= 1, 0 <= y <= 1 and 0 <= z <= x using a triple integral.

    Answer: ______________

  4. What is the triple integral of 1 over the unit cube with 0 <= x, y, z <= 1?

    Answer: ______________

  5. For a circular cylindrical tank centred on the z-axis with flat horizontal ends, Tom says spherical coordinates give constant independent radius, angle and height bounds.

    Circle one:   True   False

  6. Evaluate the spherical volume integral for a ball of radius 3 centred at the origin, with phi measured from the positive z-axis.

    • 36pi
    • 12pi
    • 18pi

  7. Density is x² + y² + z². With spherical phi measured from the positive z-axis, what coefficient multiplies d rho d phi d theta in the mass integral?

    • rho³ sin(phi)
    • rho² sin(phi)
    • rho⁴ sin(phi)
    • rho⁴ cos(phi)

  8. In spherical coordinates, phi is measured from the positive z-axis and theta around that axis. What is dV?

    • rho² cos(phi) d rho d phi d theta
    • rho² d rho d phi d theta
    • rho² sin(phi) d rho d phi d theta

  9. You set up a triple integral with z integrated first. Where do the outer two limits and the inner z-limits come from?

    • All three from the integrand; bounding surfaces do not affect the limits
    • Outer two from the plane projection, inner from lower and upper surfaces
    • Outer two from lower and upper surfaces, inner from the plane projection

  10. A triple integral intended to calculate a ball's volume disagrees with its known volume. What should you check?

    • The integrand, Jacobian, limits and arithmetic
    • Only the limits, because arithmetic cannot affect volume
    • Nothing, because a coordinate change may change volume
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Integrating solids and choosing coordinates W1-mt_RFeK8LD_jT-s1

  1. 6 · Volume is length times width times height: 2 times 1 times 3.
  2. d rho d phi d theta · Differentials follow the integration order from the innermost variable outward.
  3. 0.5 · The inner z-integral gives x, the y-integral leaves x, and the x-integral gives 1/2.
  4. 1 · Integrating 1 gives the volume, and the unit cube has volume 1.
  5. False · Those constant independent bounds are natural in cylindrical coordinates. Spherical bounds for this tank are coupled.
  6. 36pi · The radial integral gives 9, the phi integral gives 2 and the theta integral gives 2pi. Their product is 36pi.
  7. rho⁴ sin(phi) · The density becomes rho² and the volume factor is rho² sin(phi), giving rho⁴ sin(phi).
  8. rho² sin(phi) d rho d phi d theta · The local lengths multiply to rho² sin(phi) d rho d phi d theta under the stated convention.
  9. Outer two from the plane projection, inner from lower and upper surfaces · Use the plane projection for the outer two limits and the heights between the surfaces for the inner limits.
  10. The integrand, Jacobian, limits and arithmetic · A mismatch flags a problem but does not identify its location. Check both the region description and the calculation.
Worksheet · LightMySky