# Edin's question via email about volume by revolution.

#### Prove It

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MHB Math Helper
R is the region bounded above by \displaystyle \begin{align*} y = 1 + x \end{align*} and below by \displaystyle \begin{align*} y = 2\,x^2 \end{align*}.

(a) If R is rotated about the line \displaystyle \begin{align*} y = 3 \end{align*}, construct a definite integral for the volume of the region generated.

(b) If R is rotated about the line \displaystyle \begin{align*} x = -1 \end{align*} construct a definite integral for the volume of the region generated.
We should note that the two functions intersect at \displaystyle \begin{align*} x = -\frac{1}{2} \end{align*} and \displaystyle \begin{align*} x = 1 \end{align*}.

(a) Using the method of washers, the inner radius is \displaystyle \begin{align*} 3 - \left( x + 1 \right) = 2 - x \end{align*} and the outer radius is \displaystyle \begin{align*} 3 - 2\,x^2 \end{align*}, so the volume is

\displaystyle \begin{align*} V &= \int_{-\frac{1}{2}}^1{ \pi \, \left( 3 - 2\,x^2 \right) ^2\,\mathrm{d}x } - \int_{-\frac{1}{2}}^1{ \pi \, \left( 2 - x \right) ^2 \,\mathrm{d}x } \\ &= \pi \int_{-\frac{1}{2}}^1{ \left[ \left( 3 - 2\,x^2 \right) ^2 - \left( 2 - x \right) ^2 \right] \,\mathrm{d}x } \\ &= \pi \int_{-\frac{1}{2}}^1{ \left[ 9 - 12\,x^2 + 4\,x^4 - \left( 4 - 4\,x + x^2 \right) \right]\,\mathrm{d}x } \\ &= \pi \int_{-\frac{1}{2}}^1{ \left( 4\,x^4 - 13\,x^2 + 4\,x + 5 \right) \,\mathrm{d}x } \end{align*}

(b) Using the method of cylindrical shells, the radius of each cylinder is \displaystyle \begin{align*} x + 1 \end{align*} and the height of each cylinder is \displaystyle \begin{align*} x + 1 - 2\,x^2 \end{align*}. The area of each rectangular cylindrical shell is \displaystyle \begin{align*} 2\,\pi\,r\,h = 2\,\pi\,\left( x + 1 \right) \left( x + 1 - 2\,x^2 \right) \end{align*}, and they will all be summed up between \displaystyle \begin{align*} x = -\frac{1}{2} \end{align*} and \displaystyle \begin{align*} x = 1 \end{align*}, thus the volume is

\displaystyle \begin{align*} V &= \int_{-\frac{1}{2}}^1{ 2\,\pi\,\left( x + 1 \right) \left( x + 1 - 2\,x^2 \right) \,\mathrm{d}x } \\ &= 2\,\pi \int_{-\frac{1}{2}}^1{ \left( x^2 + x - 2\,x^3 + x + 1 - 2\,x^2 \right)\,\mathrm{d}x } \\ &= 2\,\pi \int_{-\frac{1}{2}}^1{ \left( 2\,x + 1 - x^2 - 2\,x^3 \right) \,\mathrm{d}x } \end{align*}