calculus-III-evaluate-0-pi-2-0-x-cos-y-pi-2-x-pi-2-y-dydx-

Question Number 138114 by mnjuly1970 last updated on 10/Apr/21

\dots \dots c a l c u l u s \dots . . (I I I) \dots \dots

e v a l u a t e ::

ϕ \overset{? ? ?}{=} \int_{0}^{\frac{π}{2}} \int_{0}^{x} \frac{c o s (y)}{\sqrt{(\frac{π}{2} - x) (\frac{π}{2} - y)}} d y d x

Commented by Kamel last updated on 10/Apr/21

Commented by mnjuly1970 last updated on 10/Apr/21

t h a n k s a l o t m r . . k a m e l

Answered by mnjuly1970 last updated on 10/Apr/21

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

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

= 2 \int_{0}^{\frac{π}{2}} c o s (y) d y = 2 s i n (\frac{π}{2}) = 2 \dots ✓