I have to determine whether the following improper integral is convergent or divergent: the volume obtained when the area between the $x$-axis for $x\ge 1$ and the graph of $f(x)=\frac{\sin(x)}{x}$ is revolved about the $x$-axis.
What should I do in this question? Evaluate limit for integral $\frac{\sin(x)}{x}$ from $1$ to $+\infty$ or what?
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By Cavalieri's principle, we just have to understand if the integral: $$ V=\pi\int_{0}^{+\infty}\frac{\sin^2 x}{x^2}\,dx \tag{1}$$ is convergent or not. Obviously it is, since $0\leq\sin^2 x\leq x^2$ gives: $$\int_{0}^{+\infty}\frac{\sin^2 x}{x^2}\,dx \leq \int_{0}^{1}1\,dx + \int_{1}^{+\infty}\frac{dx}{x^2} = 2.\tag{2}$$ Using integration by parts, it is not difficult to compute $V$:
$$ \color{red}{V}=\pi\int_{0}^{+\infty}\frac{\sin^2(x)}{x^2}\,dx = \pi\int_{0}^{+\infty}\frac{\sin(2x)}{x}\,dx = \pi\int_{0}^{+\infty}\frac{\sin(t)}{t}\,dt = \color{red}{\frac{\pi^2}{2}}.\tag{3}$$
The resulting solid is composed by infinitely many "pieces" which meet in single points (as the figure shows clearly). You can put the $k$-th piece in a cylinder which has radius which is $C/k$ and height which is $\pi$. So the $k$-th cylinder has volume $C'/k^2$... and the sum is then finite.