Question

Set up a double integral in rectangular coordinates for calculating the volume of the solid under the graph of the function f(x,y) = 32 - x^2 - y^2 and above the plane z = 7. . . (include limits of integration). .

Answer 1

  Region of integration: 32 - x^2 - y^2 = 7. 
==> x^2 + y^2 = 25, a circle. 

So, the volume equals 
∫∫ [(32 - x^2 - y^2) - 7] dA 
= ∫(r = 0 to 6) ∫(θ = 0 to 2π) (25 - r^2) * (r dθ dr) 
= ∫(r = 0 to 6) 2π (25r - r^3) dr 
= 2π (r^2 - r^4/4) {for r = 0 to 6} 
= 648π.