Applications – Volumes of Revolution

Problem 242: Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

(1) $\begin{equation*} y = 3x^2, \qquad x = 1, y = 0, \end{equation*}$

about the $x$ -axis.

Solution: The volume of revolution is given by

(2) $\begin{equation*} V = \pi \int_{a}^{b}[R(x)]^2 \ dx,\end{equation*}$

where $R(x)$ is the distance between the two curves. We are given one limit of integration $x=1$ . We need to find the other one. That is,

(3) $\begin{equation*}y = 3x^2 \implies 0 = 3x^2 \implies x = 0. \end{equation*}$

We now have

(4) $\begin{align*} V & = \pi \int_{a}^{b}[R(x)]^2 \ dx \\ & = \pi \int_{0}^{1}[3x^2 - 0]^2 \ dx \\ & = \pi \frac{9x^2}{5}\bigg|_{0}^{1} \ dx \\ & = \frac{9}{5} \pi \\ & \approx 5.65486677646.\end{align*}$

That is,

(5) $\begin{equation*} V = \frac{9}{5} \pi. \end{equation*}$

One Problem at a Time

Applications – Volumes of Revolution

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