Question

Use cylindrical shells to find the volume of the solid that results when the region enclosed by x = y^2 and x = y is revolved about the line y= -1

Answer 1

Horizontal axis of revolution

$$V=2\pi\displaystyle\int_{c}^dp(y)h(y)dy$$

$y^2=y=>y_1=0, y_2=1$

$p(y)=y-(-1)=y+1$

$h(y)=y-y^2$

$$V=2\pi\displaystyle\int_{0}^1(y+1)(y-y^2)dy=$$

$$=2\pi\displaystyle\int_{0}^1(y^2-y^3+y-y^2)dy=$$

$$=2\pi\big[\dfrac{y^2}{2}-\dfrac{y^4}{4}\big]\begin{matrix} 1 \\ 0 \end{matrix}=\dfrac{1}{2}\pi\ (units^3)$$