How do you even integrate an ellipse? The question is…
Rotating the ellipse $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ about the $x$-axis generates an ellipsoid. Compute its volume. The way i know how to rotate a function around the $x$ axis is by multiplying the integrand of $f(x)^2$ by $\pi$
$$\text{volume} = \pi \int f(x)^2$$
Best Answer
Volume of revolution of a curve about the x-axis is calculated by
$$V= \int A(x)\,dx $$
where $A(x)$ is the cross sectional area
$$A(x)= \pi y^2$$
$$dV=\pi y^2\,dx$$ substituting $y^2=b^2(1-{x^2\over a^2})$ we get,
$$V=\int_{-a}^{a} \pi b^2(1-{x^2\over a^2})dx$$
On integrating we get $$V=\frac{4}{3}\pi ab^2$$