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Question Number 141855 by rs4089 last updated on 24/May/21

Answered by TheSupreme last updated on 24/May/21

$$E=\{(x,y)\mid y<x<a\vee 0<y<a\}$$$$E^1=\{(\rho ,\theta )\mid 0<\rho <a,0<\theta <\frac{\pi }{4}\}$$$$det\mid J\mid =\rho$$$${\int }_0^a{\int }_0^{\frac{\pi }{4}}\frac{{\rho }^3{cos}^2\left(\theta \right)}{{\rho }^3}d\rho d\theta =a{\int }_0^{\frac{\pi }{4}}{cos}^2\left(\theta \right)d\theta$$$$\int {cos}^2\left(\theta \right)d\theta =\int \frac{1}{2}+\frac{1}{2}(2{cos}^2\theta -1)d\theta =\frac{\theta }{2}+\frac{1}{2}\int cos\left(2\theta \right)d\theta =2\theta +\frac{1}{4}sin\left(2\theta \right)$$$$...=a[\frac{\pi }{8}+\frac{1}{4}]$$

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