Change the Cartesian Integral into an equivalent polar integral then evaluate the polar integral.

Given Cartesian Integral: ∫0-1∫0-sqrt (1-y^2) (x^2 + y^2) dxdy

the ingegral I=π/4

Step-by-step explanation:

From the integral

I=∫0-1∫0-sqrt (1-y^2) (x^2 + y^2) dxdy

from polar coordinates

r²=x² + y²

then for r=1

√ (1 - y²) = x

for y=1 → x=0, for y=0 → x=1

then the integration area is a hemisphere or radius r=1

therefore

I = ∫0-1∫0-sqrt (1-y^2) (x^2 + y^2) dxdy = (1/2) ∫2π-0 ∫1-0 r² * r drdθ (the additional r is due to the Jacobian of the transformation to polar coordinates, 1/2 because is an hemisphere so it would be the half of the value of the total sphere)

I = (1/2) ∫2π-0 ∫1-0 r³drdθ = (1/2) ∫2π-0 (1⁴/4 - 0⁴/4) dθ = (1/2) ∫2π-0 (1/4) * dθ = (1/2) * (1/4) * 2π = π/4