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A town hall has a roof shaped like a hemisphere of radius 5 m, standing on a flat floor…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

A town hall has a roof shaped like a hemisphere of radius $5$ m, standing on a flat floor. A builder needs the volume of air under the roof to size the heating. To find it, turn the quarter circle $y = \sqrt{25 - x^2}$ , for $0 \le x \le 5$ , about the $x$ -axis. Here $x$ and $y$ are in metres.

Need a reminder? The key ideaShow

For a region turned about the $x$ -axis, a slice is a disc of volume $\pi y^2\,dx$ , so $V = \pi\displaystyle\int_a^b y^2\,dx$ . With a hole, use washers: $V = \pi\displaystyle\int_a^b (R^2 - r^2)\,dx$ . About the $y$ -axis, use $\pi x^2\,dy$ and write $x^2$ in terms of $y$ .

The worked solution

Have you tried it?

Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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