Question

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

Answer 1

Answer:

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