題目

5. (10 points) Find the volume of the solid generated by rotating the curve

y=\sin(x^2)

over $0 \le x \le 1$ around the $y$ -axis.

解答

展開

半徑為 $x$ 、高為 $\sin(x^2)$ 的薄柱殼體積為 $2\pi x\sin(x^2)\,\mathrm{d}x$ ，故

\begin{align*} V =&\,2\pi\int_0^1x\sin(x^2)\,\mathrm{d}x\\[4mm] =&\,\pi\int_0^1\sin u\,\mathrm{d}u\\[4mm] =&\,\boxed{\pi(1-\cos1)}. \end{align*}