Single Variable Calculus, Chapter 6, 6.3, Section 6.3, Problem 4

Use the shell method to find the volume generated by rotating the region bounded by the curves $y = x^2, y = 0, x = 1$ about the $y$-axis. Sketch the region and a typical shell








If we use a vertical strips, notice that the distance of these strips from the $y$-axis is $x$. If we revolve this length about the $y$-axis, you'll get the circumference $C = 2 \pi x$. Also, notice that the height of the strips resembles the height of the cylinder as $\displaystyle H = y_{\text{upper}} - y_{\text{lower}} = x^2 - 0$. Thus, we have..



$
\begin{equation}
\begin{aligned}

V =& \int^b_a C(x) H(x) dx
\\
\\
V =& \int^1_0 2 \pi x (x^2) dx
\\
\\
V =& 2 \pi \int^1_0 x^3 dx
\\
\\
V =& 2 \pi \left[ \frac{x^4}{4} \right]^1_0
\\
\\
V =& 2 \pi \left[ \frac{(1)^4}{4} - \frac{(0)^4}{4} \right]
\\
\\
V =& \frac{\pi}{2} \text{ cubic units}

\end{aligned}
\end{equation}
$