Answer
$\dfrac{4\pi a^2 b}{3}$
Work Step by Step
The shell method to compute the volume of a region: The volume of a solid obtained by rotating the region under $y=f(x)$ over an interval $[m,n]$ about the y-axis is given by:
$V=2 \pi \int_{m}^{n} (Radius) \times (height \ of \ the \ shell) \ dy=2 \pi \int_{m}^{n} (y) \times f(y) \ dy$
Now, $V=2\pi \int_{-b}^{b} (\sqrt{a^2-\dfrac{a^2y^2}{b^2}})^2 \ dy\\= \pi \int_{-b}^{b} (a^2-\dfrac{a^2y^2}{b^2}) \ dy \\= \pi [a^2y -\dfrac{a^2 y^3}{3b^2}]_{-b}^{b} \\=\pi (a^2b - \dfrac{a^2 b^3}{3b^2 }+a^2b -\dfrac{a^2 b^3}{3b^2}) \\= \pi [2a^2 b -\dfrac{a^2 b}{3}-\dfrac{2 a^2 b}{3}] \\=\dfrac{4\pi a^2 b}{3}$
