Calculus double integrals

Question

Find the volume of the solid bounded by the paraboloids  and 

Answer 1

Lets find The projection of the intersection of the two solids on the xy- plane.

4 + x² + y² = 9 −2x² −2y²

3 x² +3 y² = 5

x² + y² = 5/3 is a circle centered on the origin

with radius √(5/3) since

The using cylindrical coordinates

z₁ = 4 + r²

z₂ = 9 −2 r²

V= ∫ ^θ=2π _θ=0 ∫ ^{r= √(5/3)}_r=0[5 −3r²]rdrdθ= ∫ ^θ=2π _θ=0 ∫ ^{r= √(5/3)}_r=0[5r −3r³]drdθ =

∫ ^θ=2π _θ=0 [5r²/2 −3r⁴/4]^{r= √(5/3)}_r=0dθ = ∫ ^θ=2π _θ=0[25/12]dθ =25π/6