Find an expression, but do not evaluate an integral for the volume of the solid obtained by rotating the region bounded by the curves $y = x, y = 4x - x^2$ about $x = 7$.

By using vertical strips, notice that the distance of the strips from the line $x = 7$ is $7 - x$. If you revolve this distance about $x = 7$, you'll get a circumference of $C = 2 \pi (7 - x)$. Also, notice that the height of the strips resembles the height of the cylinder as $H = y_{\text{upper}} - y_{\text{lower}} = 4x - x^2 - (x)$. Thus, we have

$\displaystyle V = \int^b_a C(x) H(x) dx$

In order to get the values of the upper and lower limits, we simply determine the points of intersection of the curves..

$ \begin{equation} \begin{aligned} & x = 4x - x^2 \\ & x^2 - 3x = 0 \\ & x(x - 3) = 0 \\ & \text{We get,} \\ & x = 0 \text{ and } x = 3 \end{aligned} \end{equation} $

Therefore, the expression for the volume is..

$ \begin{equation} \begin{aligned} & V = \int^3_0 2 \pi (7 - x)(4x - x^2 - x) dx \\ \\ & V = 2 \pi \int^3_0 (21 x - 7x^2 - 3x^2 + x^3) dx \\ \\ & V = 2 \pi \int^3_0 (21x - 10x^2 + x^3) dx \end{aligned} \end{equation} $

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