Evaluate-c-ydy-where-c-is-a-circle-x-2-y-2-4-

Question Number 135872 by Engr_Jidda last updated on 16/Mar/21

E v a l u a t e \oint_{c} y d y w h e r e c i s a c i r c l e x^{2} + y^{2} = 4

Answered by mathmax by abdo last updated on 16/Mar/21

y = ξ \sqrt{4 - x^{2}} \Rightarrow \int_{C} ydy = \int_{- 2}^{2} ξ \sqrt{4 - x^{2}} ξ \frac{- 2 x}{2 \sqrt{4 - x^{2}}} dx

= \int_{- 2}^{2} xdx = 0 (ξ^{2} = 1) and

\int_{C} ydx = \int_{- 2}^{2} ξ \sqrt{4 - x^{2}} dx =_{x = 2 sint} ξ \int_{- \frac{π}{2}}^{\frac{π}{2}} 2 cost (2 cost) dt

= 4 ξ \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{1 + \cos (2 t)}{2} dt = 2 ξ \int_{- \frac{π}{2}}^{\frac{π}{2}} (1 + \cos (2 t)) dt

= 2 π ξ + ξ {[\sin (2 t)]}_{- \frac{π}{2}}^{\frac{π}{2}} = 2 π ξ

Commented by Engr_Jidda last updated on 17/Mar/21

t h a n k y o u s o m u c h s i r .

Commented by mathmax by abdo last updated on 17/Mar/21

you are welcome