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Integrating solids and choosing coordinates

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: ______________
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
Find the volume of 0 <= x <= 1, 0 <= y <= 1 and 0 <= z <= x using a triple integral.
Answer: ______________
What is the triple integral of 1 over the unit cube with 0 <= x, y, z <= 1?
Answer: ______________
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
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
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)
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
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
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

Answer key

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

Integrating solids and choosing coordinates W1-mt_RFeK8LD_jT-s1

6 · Volume is length times width times height: 2 times 1 times 3.
d rho d phi d theta · Differentials follow the integration order from the innermost variable outward.
0.5 · The inner z-integral gives x, the y-integral leaves x, and the x-integral gives 1/2.
1 · Integrating 1 gives the volume, and the unit cube has volume 1.
False · Those constant independent bounds are natural in cylindrical coordinates. Spherical bounds for this tank are coupled.
36pi · The radial integral gives 9, the phi integral gives 2 and the theta integral gives 2pi. Their product is 36pi.
rho⁴ sin(phi) · The density becomes rho² and the volume factor is rho² sin(phi), giving rho⁴ sin(phi).
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.
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.
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.