Question

The volume of the solid of revolution formed by rotating the finite region bounded by the x-axis and the graph of $y=1-x^2$, about the x-axis is $\frac{a\pi }{15}$. What will be the value of a?

Answer 1

$\Rightarrow$ ${\int }_{-1}^1\pi y^{2}dx$

$\Rightarrow$ $\pi {\int }_{-1}^1[1-x^2{]}^{2}dx$

$\Rightarrow$ $2\pi {\int }_0^1[1-x^2{]}^{2}dx$ (even function)

$\Rightarrow$ $2\pi \times \frac{8}{15}$

$\Rightarrow$ $\frac{16\pi }{15}$