Solid of Revolution About a Line Other Than an Axis

If a curve  
, defined between points  
\[x_1, \; x_2\]
  is rotated about the  
  axis, it forms a solid , called a volume or solid of revolution. The volume can be found by integration.
\[V= \pi \int{x_2}_{x_1} y^2 dx \]

What though if the curve is rotated about some line other than an axis?
If the curve is rotated about a line parallel to the  
  axis, we can write a volume element as  
\[dV= \pi r^2dy\]
Suppose the curve  
\[y=x^2, \; 0 \lt x \lt 1\]
  is rotated about the line  
. What is the volume of the solid formed?
We can write  
\[r=1-\sqrt{y}, \; 0 \lt y \lt 1\]
  then the volume element becomes  
\[dV=(1- \sqrt{y})^2dy\]
. The solid of revolution has volume
\[\begin{equation} \begin{aligned} V &= \pi \int^1_0 (1- \sqrt{y})^2dy \\ &= \int^1_0 (1-2 \sqrt{y}+y)dy \\ &= [y-\frac{2 \sqrt{y}}{3} + \frac{y^2}{2} ]^1_0 \\ &= (1- \frac{2}{3} + \frac{1}{2} )-(0) \\ &= \frac{5}{6}\end{aligned} \end{equation} \]

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