calculate-D-x-2-y-2-dxdy-D-x-y-R-2-1-2-x-2-y-2-3-and-y-0-

Question Number 78705 by abdomathmax last updated on 20/Jan/20

c a l c u l a t e \int \int_{D} (x^{2} + y^{2}) d x d y

D = {(x, y) \in R^{2} / \frac{1}{2} ⩽ x^{2} + y^{2} ⩽ 3 a n d y ⩾ 0}

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

x = rcos (θ), y = rsin (θ) \Rightarrow dxdy = rdrd θ

D \Leftrightarrow \frac{1}{2} ⩽ r^{2} ⩽ 3 \bar{} \sin (θ) ⩾ 0 \Rightarrow θ \in [0, π]

r \in [\frac{1}{\sqrt{2}}, \sqrt{3}]

\int \int_{D} (x^{2} + y^{2}) dxdy = \int_{0}^{π} \int_{\frac{1}{\sqrt{2}}}^{\sqrt{3}} r^{3} drd θ = π . {[\frac{r^{4}}{4}]}_{\frac{1}{\sqrt{2}}}^{\sqrt{3}}

= π (\frac{9}{4} - \frac{1}{16}) = \frac{35 π}{16}