Question

The base of a solid in the xy-plane is the circle x^2 + y^2 = 16. Cross sections of the solid perpendicular to the y-axis are semicircles. What is the volume, in cubic units, of the solid?

Answer 1

I got 2pi/3

Answer 2

This solid is exactly a hemisphere of radius \(4\).
\[V=\frac{1}{2}\left(\frac{4}{3}\pi (4^2)\right)=\frac{32\pi}{3}\]

Answer 3

ohh ok tthtanks :)

Answer 4

Whoops, I meant \(\color{red}{(4^3)}\), should be \(\dfrac{128\pi}{3}\).

Answer 5

Are you supposed to find the volume using integration?

Answer 6

Im not sure

Answer 7

I think the answer is 32π/3

Answer 8

Why do you say that?

Answer 9

never mind tht does seem right

Answer 10

Right. In case you need more convincing (and judging by the fact that you seem to have posted several calc question recently), here's a justification using integration.

Each section's radius is given by the vertical distance to the horizontal axis, which is given by the positive root \(\sqrt{16-x^2}\). The area of any one cross section is \(\dfrac{\pi r(x)^2}{2}\), with \(r(x)\) the radius which depends on the section's position along the axis.

The total volume is the sum of infinitely many, infinitesimally thin sections, so you have
\[V=\int_{-4}^4 \frac{\pi(\sqrt{16-x^2})^2}{2}\,\mathrm{d}x\]which you'll find to coincide with \(\dfrac{128\pi}{3}\).