Question

Evaluate the following integral over the specified region. Assume (r comma theta )(r,θ) are polar coordinates. ModifyingBelow Integral from nothing to nothing Integral from nothing to nothing With Upper R StartFraction dA Over (16 plus x squared plus y squared )squared EndFraction∫∫R dA 16+x2+y22​; Upper R equals StartSet (r comma theta ): 1 less than or equals r less than or equals 3 comma 0 less than or equals theta less than or equals pi EndSetR={(r,θ): 1≤r≤3, 0≤θ≤π} M

Answer 1

Explanation:

We have to integrate the function

`16+x^2+y^2` over the region

`1\leq r\leq 3\\0\leq \theta \leq \pi`

Let us convert into polar coordinates

x=cost : y = sint

Then we get dxdy = rdr dt

So we get the integral as

`\int \int _R (16+x^2+y^2)^2 dxdy\\= \int\limits^3_0 \int\limits^\pi_0( {16+r^2}^2) (r)drdt\\ =(t) (((16+r^2)^3)/(6)`

Substitute the limits to get

Answer

=
`(\pi-0) [(1)/(6) (25-16)\\=1.5 \pi`