a Evaluate the volume of the solid bounded by the paraboloid

a) Evaluate the volume of the solid bounded by the paraboloids z = 5x^2 + 5y^2
and z = 6 -7x^2 -y2 and

b) the volume of the solid bounded by the paraboloid z = 2 - (x^2 + y^2) and
the cone z =sqrt(x^2 + y^2). Hint. To find the intersection of the surfaces in b), you should
let sqrt(x^2 + y^2) = t..

Solution

1) the paraboloids z = 5x^2 + 5y^2

and z = 6 -7x^2 -y2

=> 5x^2 + 5y^2 =6 -7x^2 -y2

which simplifies to  => 2 x^2 + y^2 = 1

This circle and its interior constitute the base of the solid. The volume V is given by

=> ?( 12 x^2 + 6 y^2 -6 ) da

=> where the integral is taken over the region bounded by 2x