What-will-be-the-vallu-of-a-a-x-2-y-dx-Where-x-2-y-2-a-2-and-y-0-

Question Number 17206 by Arnab Maiti last updated on 02/Jul/17

What will be the vallu of \int_{- a}^{a} x^{2} y dx ?

Where x^{2} + y^{2} = a^{2} and y ⩾ 0

Answered by mrW1 last updated on 02/Jul/17

let x = a \cos θ

and y = a \sin θ

dx = - asin θ d θ

x \in [- a, a] and y ⩾ 0 \equiv θ \in [π, 0]

\int_{- a}^{a} x^{2} ydx = \int_{π}^{0} a^{2} \cos^{2} θ a \sin θ (- asin θ) d θ

= - a^{4} \int_{π}^{0} \cos^{2} θ \sin^{2} θ d θ

= a^{4} \int_{0}^{π} \cos^{2} θ \sin^{2} θ d θ

= \frac{a^{4}}{4} \int_{0}^{π} {(2 \cos θ \sin θ)}^{2} d θ

= \frac{a^{4}}{4} \int_{0}^{π} \sin^{2} 2 θ d θ

= \frac{a^{4}}{8} \int_{0}^{π} [1 - \cos 4 θ] d θ

= \frac{a^{4}}{8} {[θ - \frac{1}{4} \sin 4 θ]}_{0}^{π}

= \frac{a^{4}}{8} \times π

= \frac{π a^{4}}{8}

Commented by Arnab Maiti last updated on 02/Jul/17

E \times c {ille}^{nt}!!