find-the-area-of-the-region-bounded-by-the-semicircle-y-a-2-x-2-and-the-x-a-and-the-line-y-a-by-using-intigiral-pleas-sir-help-me-

Question Number 72884 by mhmd last updated on 04/Nov/19

f i n d t h e a r e a o f t h e r e g i o n b o u n d e d b y t h e s e m i c i r c l e y = \sqrt{a^{2} - x^{2}} a n d t h e x = + - a a n d t h e l i n e y = - a ? b y u s i n g i n t i g i r a l

p l e a s s i r h e l p m e

Commented by kaivan.ahmadi last updated on 04/Nov/19

\int_{- a}^{a} (\sqrt{a^{2} - x^{2}} + a) d x =

x = a s i n θ \Rightarrow d x = a c o s θ d θ

\sqrt{a^{2} - x^{2}} = a c o s θ

{\begin{cases} x = a \Rightarrow θ = \frac{π}{2} \\ x = - a \Rightarrow θ = - \frac{π}{2} \end{cases}

\int_{- \frac{π}{2}}^{\frac{π}{2}} a^{2} (c o s θ + {c o s}^{2} θ) d θ = a^{2} (s i n θ + \frac{θ + \frac{1}{2} s i n 2 θ}{2}) ∣_{- \frac{π}{2}}^{\frac{π}{2}} =

a^{2} [(1 + \frac{π}{4}) - (- 1 - \frac{π}{4})] = a^{2} (2 + \frac{π}{2})