Question

In the following exercises, evaluate the double integral ∫Rf(x,y)dA over the polar rectangular region D.

f(x,y)=3 √x²+y ²
where D={(r,θ)∣0≤r≤2,3π≤θ≤π}
Include a drawing of the region of integration.

Answer 1

`-16\pi`

Explanation:

`\displaystyle \iint_Rf(x,y)\,dA\\\\=\iint_Df(r\cos\theta,r\sin\theta)\,r\,dr\,d\theta\\\\=\iint_D3√(r^2\cos^2\theta+r^2\sin^2\theta)\,r\,dr\,d\theta\\\\=\iint_D3r^2\,dr\,d\theta\\\\=\int^\pi_(3\pi)\int^2_03r^2\,dr\,d\theta\\\\=\int^\pi_(3\pi)8\,d\theta\\\\=8\pi-8(3\pi)\\\\=8\pi-24\pi\\\\=-16\pi`