Question

Evaluate the integral by converting to polar coordinates`\int_{0}^{sqrt3} \int_{y}^{sqrt(4-y^2)} (dxdy)/(4+x^(2)+y^(2))`.?

Answer 1

`int_0^sqrt2 int_y^sqrt(4-y^2)(dxdy)/(4+x^2+y^2)=1/8piln2`

Explanation:

I will assume that the limits given in the question are wrong and that the actual integral is `int_0^sqrt2 int_y^sqrt(4-y^2)(dxdy)/(4+x^2+y^2)`

We are integrating the area between a circle of radius `4` centred at the origin and the function `y=x`.

We use `x=rcostheta,y=rsintheta`. The Jacobian `J(r,theta)=del(x,y)"/"del(r,theta)=r` so `dxdy=rdrd theta`. The limits are `0<=r<=2` and `pi"/"4<=theta<=pi"/"2`. Hence, we have

`int_0^sqrt2 int_y^sqrt(4-y^2)(dxdy)/(4+x^2+y^2)=int_(pi/4)^(pi/2)int_0^2 r/(4+r^2) drd theta=int_(pi/4)^(pi/2) [1/2ln(4+r^2)]_0^2 d theta=int_(pi/4)^(pi/2)1/2ln2d theta=1/8piln2`

A task for you would be to fill in the gaps above.