Question

determine the mass of the lamina corresponding to the first quadrant portion of the circle x^2+y^2 =25 where the density at thr point of (x,y) is f(x,y)=k*sqrt(x^2+y^2)

Answer 1

$\iint_{x^2 + y^2 \leq 25, \ x \geq 0, \ y \geq 0} k \sqrt{x^2 + y^2} \,dx\,dy =$

$= \int_{0}^{5} \,dx \int_{0}^{\sqrt{25 -x^2}} k \sqrt{x^2 + y^2} \,dy =$

$\begin{cases} x = rcos\phi \\ y = rsin\phi\\ \end{cases} \\$

$= \int_{0}^{\pi/2} \,d\phi \int_{0}^{5} k \sqrt{(rcos\phi)^2 + (rsin\phi)^2} \,rdr =$

$= \int_{0}^{\pi/2} \,d\phi \int_{0}^{5} kr^2 \,dr = \int_{0}^{\pi/2} k *5^3/3 \,d\phi = k *5^3/3 * \pi/2 = 125\pi k/6$