Find the volume of the solid bounded by the paraboloid z8x23

Find the volume of the solid bounded by the paraboloid z=8-x²-3y² and the hyperbolic paraboloid z=x²-y²

Solution

Given: paraboloid z=8-x²-3y² and the hyperbolic paraboloid z=x²-y²

           Curve of intersection:
x^2 -y^2 = 8 - x^2 - 3y^2

    2x^2 + 2y^2 = 8,

     x^2 + y^2 = 4,
==> x^2 + y^2 = 4, it is a circle.

Using polar coordinates, the volume equals
[(8 - x^2 - 3y^2) - (x^2 - y^2)] dA
= ( = 0 to 2) (r = 0 to 2) [(8 - r^2) - r^2] * r dr d
= ( = 0 to 2) (r = 0 to 2) (8r - 2r^3) dr d
= 2 * (4r^2 - r^4/2) {for r = 0 to 2}
= 16.

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