Question

Evaluate the integral ∬D​(xy)dA, where D is the intersection of D1​={(r,θ)∣r≤4cosθ} and D2​={(r,θ)∣r≤4sinθ}.

Answer 1

4

Explanation:

Let
`x=r\cos\theta`,
`y=r\sin\theta`, and
`dA=r\,dr\,d\theta`:

`\displaystyle \iint_Dxy\,dA\\\\=\iint_Dr^3\sin\theta\cos\theta\,dr\,d\theta\\\\=\int^(\pi)/(4)_0\int^\sqrt8}_0r^3\sin\theta\cos\theta\,dr\,d\theta\\\\=\int^(\pi)/(4)_0\sin\theta\cos\theta\int^\sqrt8}_0r^3\,dr\,d\theta\\\\=\int^(\pi)/(4)_0\sin\theta\cos\theta\biggr(((√(8))^4)/(4)\biggr)\,d\theta\\\\=\int^(\pi)/(4)_016\sin\theta\cos\theta\,d\theta`

Let
`u=\sin\theta` and
`du=\cos\theta\,d\theta` and the bounds be from
`u_1=\sin0=0` to
`u_2=\sin((\pi)/(4))=(√(2))/(2)`:

`\displaystyle =\int^(√(2))/(2)_016u\,du\\\\=8u^2\biggr|^(√(2))/(2)_0\\\\=8\biggr((√(2))/(2)\biggr)^2\\\\=8\biggr((2)/(4)\biggr)\\\\=4`