D-4y-2-sin-xy-dxdy-D-x-y-x-0-y-pi-2-0-x-y-0-y-pi-2-

Question Number 205935 by EJJDJX last updated on 03/Apr/24

\int \int_{D} (4 y^{2} s i n (x y)) d x d y = ? ? ?

D : x = y x = 0 y = \sqrt{\frac{π}{2}}

0 ⩽ x ⩽ y 0 ⩽ y ⩽ \sqrt{\frac{π}{2}}

Answered by Berbere last updated on 03/Apr/24

\int_{0}^{\sqrt{\frac{π}{2}}} \int_{0}^{y} 4 y^{2} s i n (x y) d x d y

= \int_{0}^{\sqrt{\frac{π}{2}}} - 4 y {[c o s (x y)]}_{0}^{y} d y

= \int_{0}^{\sqrt{\frac{π}{2}}} - 4 y [c o s (y^{2}) - 1) d y

= - 2 \int_{0}^{\sqrt{\frac{π}{2}}} (c o s (y^{2}) - 1) {d y}^{2}

= 2 \int_{0}^{\sqrt{\frac{π}{2}}} (s i n (y^{2}) - y^{2})

= 2 (1 - \frac{π}{2})

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