MediumChange of variables

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §14.4 Cylindrical and Spherical Coordinates, #21 · Multivariable Calculus · Change of variables

Example 5 gave the volume integral for a sphere in … coordinates

Example 5 gave the volume integral for a sphere in $r θ z$ coordinates. What is the area of the circular slice at height $z$ ? What is the area of the cylindrical shell at radius $r$ ? Integrate over slices $(d z)$ and over shells $(d r)$ to reach $\frac{4 π R ^{3}}{3}$ .

This exercise builds on another exercise in its source book, which isn't shown here.

~8 minWritten answer · check your work with Lemma

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §14.4 Cylindrical and Spherical Coordinates, #21

Stuck? A gentle nudge

No hints for this one.

Something off?

Workspace

Your working

Check your full solution

Attach photos of your pages (up to 3) or one PDF — or check what you typed above. Lemma reads the work and verifies the reasoning, not just the answer.

Your first check today doesn't need an account.

Your photos and PDFs are read once and not stored — only the transcription is kept.