Volume of solid generated by rotating $\large\frac{x^2}{a^2}+\large\frac{y^2}{b^2}$$=1$ about the major axis (x-axis since a>b) is

$V=\pi\int\limits_{-a}^a y^2 dx=2\pi\int\limits _0^a y^2 dx$

$\qquad=2\pi\int\limits_0^a b^2(1-\large\frac{x^2}{a^2})$$dx$

$\qquad=2\pi[b^2x-\large\frac{b^2}{a^2}.\frac{x^3}{3}\bigg]_0^a$

$\qquad=2\pi\bigg[ab^2-\large\frac{b^2a^3}{a^23}\bigg]$

$\qquad=\large\frac{4\pi}{3}$$ab^2$$units$