Find the volume of the solid

Question

Find the volume of the solid bounded by the plane z = 5 and the hyperboloid z = SquareRoot(9+x^2+y^2) 

Answer 1

The easiest approach to to replace ∫ dx dy with  2pi ∫ r dr  and x² +y² with r²   (polar coordinates)

 

V =  2 pi ∫ dr r (9 + r² )^1/2   The range of integration is from 0 to 4

The anti-derivative is 2 pi (1/3) (9 + r² )^3/2

 

So  V =  2 pi (1/3) [ 25^3/2  - 9^3/2  ]  =  2 pi (1/3) [ 125 - 27 ]  =  pi  196/ 3