Use polar coordinates to find the volume of the given solid.
Answer 1
Using polar coordinates,
z = 7 + 2x^2 + 2y^2 = 7 + 2r^2.
This one cuts the plane when 7 + 2r^2 = 13
==> r^2 = 3
==> r = √3, a circle.
(We take θ in [0, π/2]since we want the region in the first octant, where in particular x, y > 0.)
So the volume is equal
∫∫ (7 + 2x^2 + 2y^2) dA
= ∫(θ = 0 to π/2) ∫(r = 0 to √3) (7 + 2r^2) * (r dr dθ), conversion to polar coordinates
= ∫(θ = 0 to π/2) dθ * ∫(r = 0 to √3) (7r + 2r^3) dr
= (π/2) * (7r^2/2 + r^4/2) {for r = 0 to √3}
= 15π/2.
Hope that helps!
answer 2
kb, I’m your job, except..
when we come to this sage…
∫∫ (7 + 2x^2 + 2y^2) dA
shouldn’t it be?….
∫∫ 13 – (7 + 2x^2 + 2y^2) dA
and conversion to cylindrical coordinates.
∫∫ (6 – 2r^2)r dr dθ
= (π/2) * (6r^2/2 – r^4/2) {for r = 0 to √3}
= 9ft/4