Stack the solid, then pick round tools

Mathematics · Calculus & Analysis · ages 20-21

Name ______________________ Date ____________

What is the volume of the box with 0 <= x <= 2, 0 <= y <= 1, 0 <= z <= 3?
Answer: ______________
What is the volume element dV in cylindrical coordinates?
- r dz dr dtheta
- dz dr dtheta
- r squared sin phi dz dr dtheta
- r dr dtheta
What is the volume of the box 0 <= x <= 2, 0 <= y <= 1, 0 <= z <= 3?
Answer: ______________
What is the triple integral of 1 over the unit cube with 0 <= x, y, z <= 1?
Answer: ______________
Tom says a tall cylindrical tank is best handled in spherical coordinates.
Circle one: True False
The solid is a ball centred at the origin. Which system fits best?
- Spherical
- Cylindrical
- Rectangular
What is the volume element dV in spherical coordinates?
- rho squared sin phi d rho d phi d theta
- rho d rho d phi d theta
- r dz dr dtheta
- rho squared d rho d phi dtheta
What is the volume element dV in spherical coordinates?
- rho d rho d phi d theta
- r dz dr dtheta
- rho squared sin phi d rho d phi d theta
You set up a triple integral over a solid. Where do the outer two limits and the inner limits come from?
- All three from the integrand formula
- Outer two from the plane projection, inner from lower and upper surfaces
- Outer two from surfaces, inner from the projection
Your integral for a ball disagrees with the known volume. What does the sanity check say?
- The limits likely misdescribe the solid, so recheck the setup
- The arithmetic of 1 is broken
- Volume checks never catch errors

Answer key

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

6 · Volume is length times width times height: 2 times 1 times 3.
r dz dr dtheta · Cylindrical coordinates add the polar factor r to dz dr dtheta.
6 · Volume is length times width times height: 2 times 1 times 3 = 6.
1 · Integrating 1 gives the volume, and the unit cube has volume 1.
False · Tubes match cylindrical coordinates, while spheres match spherical ones.
Spherical · A ball is a constant rho surface, so spherical limits turn trivial.
rho squared sin phi d rho d phi d theta · The spherical Jacobian is rho squared sin phi.
rho squared sin phi d rho d phi d theta · Radial shells grow like rho squared and polar patches shrink with sin phi.
Outer two from the plane projection, inner from lower and upper surfaces · Project down for the outer pair, then read heights between surfaces for the inner one.
The limits likely misdescribe the solid, so recheck the setup · Setup is the whole battle, and a failed volume match points at the limits.