Calculus 3 double integral

Question

Find the volume of the solid bounded by the paraboloids  and 

Answer 1

The best way to solve this problem is to find the intercepts of the two curves. This is done as follows.

-7 +3x² + 3y² = 5 - 3x² -3y² this simplifies to 6x²+ 6y² =12 so x² +y² =2 and r= √2. Use polar coordinates to set up integral ∫₀^2π∫₀^√2∫_3r2_-7^5-3r2dzrdrdθ. This becomes 2π∫₀^√2(12r -6r³)dr. This becomes 2π( 6r² -3r⁴/2)₀^√2.

This then gives the volume of 12π.