Volume of Revolution Example

If a curve
$y=f)x)$
, defined between points
$x_1, \; x_2$
$x$
axis, it forms a solid , called a volume or solid of revolution. The volume can be found by integration.
$V= \pi \int{x_2}_{x_1} y^2 dx$

If the curve is defined between the points
$y_1$
and
$y_2$
, and is rotated about the
$y$
axis, the volume can be found using the formula
$V= \pi \int{y_2}_{y_1} x^2 dy$

Example: The curve
$y=x^2+x, \; 0 \lt x \lt 1$
$x$
\begin{aligned} V &= \pi \int^1_0 (x^2+x)^2dx \\ &= \int^1_0 x^4+2x^3+x^2 dx \\ &= [ \frac{x^5}{5}+ \frac{2x^4}{4}+ \frac{x}{3} ]^1_0 \\ &= (\frac{1}{5}+ \frac{1}{2} + \frac{1}{3} )-(0) \\ &= \frac{31}{30} \end{aligned}