let-a-gt-0-calculate-D-a-xdxdy-a-2-x-2-y-2-and-D-a-x-y-R-2-x-2-y-2-a-2-and-x-gt-0-

Question Number 78703 by abdomathmax last updated on 20/Jan/20

l e t a > 0 c a l c u l a t e \int \int_{D_{a}} \frac{x d x d y}{a^{2} + x^{2} + y^{2}}

a n d D_{a} = {(x, y) \in R^{2} / x^{2} + y^{2} ⩽ a^{2} a n d x > 0}

Answered by mind is power last updated on 20/Jan/20

x = rcos (θ), y = rsin (θ)

D_{a} \Leftrightarrow {(r, θ) \in R^{2} / r^{2} < a^{2} \cos (θ) > 0}

r \in [0, a], θ \in [- \frac{π}{2}, \frac{π}{2} [

\int \int_{D_{a}} \frac{xdxdy}{a^{2} + x^{2} + y^{2}} = \int_{\frac{- π}{2}}^{\frac{π}{2}} \int_{0}^{a} \frac{r^{2} \cos (θ) drd θ}{a^{2} + r^{2}}

= \int_{- \frac{π}{2}}^{\frac{π}{2}} \cos (θ) d θ . \int_{0}^{a} \frac{r^{2}}{a^{2} + r^{2}} dr

= {[\sin (θ)]}_{- \frac{π}{2}}^{\frac{π}{2}} . {[1 - \frac{a^{2}}{a^{2} + r^{2}}]}_{0}^{a}

2 {[x - aarctan (\frac{r}{a})]}_{0}^{a} 2 (a - a \frac{π}{4}) = a (\frac{4 - π}{2})