Double Integrals in Polar Coordinates

[iosa31]

Homework Statement

Use polar coordinates to find the volume of the given solid.

Enclosed by the hyperboloid -x² - y² + z² = 1 and the plane z = 2

Homework Equations

r² = x² + y², x = rcosθ, y = rsinθ

∫∫f(x,y)dA = ∫∫f(rcosθ,rsinθ)rdrdθ

The Attempt at a Solution

-x² - y² + 4 = 1 → x² + y² = 3

0 ≤ r ≤ √3
0 ≤ θ ≤ 2∏

f(x,y) = √(1 + x² + y²)
f(rcosθ,rsinθ) = √(1 + r²)

V = ∫∫r√(1 + r²)drdθ

u = 1 + r²
du = 2rdr

V = ∫∫1/2√ududθ
V = ∫1/3(u^3/2)dθ
V = 1/3∫4^3/2 - 1dθ
V = 7/3∫dθ
V = (7/3)θ = 14∏/3

 

[iosa31]

Oops, I just realized the integral should be ∫∫(2 - √(1 + r^2))rdrdθ, not ∫∫r√(1 + r^2)drdθ. Sorry about that.