What is the volume formula for a sphere of radius r?

• A. 2/3 pi r^3
• B. 4/3 pi r^3
• C. 4 pi r^3
• D. 4 pi r^2

Answer: (B)

Think about building the volume by layering thin disks from one side to the other. At a position x from the center, the slice is a circle with radius sqrt(r^2 - x^2), so its area is π(r^2 - x^2). Add up all these slices from x = -r to x = r:

V = ∫{-r}^{r} π(r^2 - x^2) dx = π[r^2x - x^3/3]{-r}^{r} = π(4r^3/3) = 4/3 π r^3.

So the volume is 4/3 π r^3. The other expressions correspond to different quantities or incorrect factors: 4 π r^2 is the surface area, not volume; and the remaining two have the wrong coefficient for the volume.