Multivariate Calculus - Bruce E. Shapiro

LECTURE 18. DOUBLE INTEGRALS IN POLAR COORDINATES 147
In the limit as ∆r and ∆θ → 0, the term ∆r/r → 0 much faster than the other
terms and hence
[
dA = lim r∆θ∆r 1 + ∆r ]
= rdrdθ
∆θ,∆r→0 2r
Hence we write
f(x, y)dxdy = f(rcosθ, rsinθ)rdrdθ
R
R
As with cartesian coordinates, we can distinguish between θ-simple and r-simple
domains in the xy-plane.
Definition 18.1 A region R in the xy plane is called θ-simple if it can be expressed
in the form
R = {(r, θ) : a ≤ r ≤ b, g(r) ≤ θ ≤ h(r)}
Double integrals over θ-simple regions have the simple form
f(x, y)dA =
R
∫ b ∫ h(r)
a
g(r)
f(r cos θ, r sin θ)dθ r dr
Definition 18.2 A region R in the xy plane is called r-simple if it can be expressed
in the form
R = {(r, θ) : α ≤ θ ≤ β, g(θ) ≤ r ≤ h(θ)}
Double integrals over θ-simple regions have the simple form
f(x, y)dA =
R
∫ β ∫ h(θ)
α
g(θ)
f(r cos θ, r sin θ)r drdθ
Figure 18.2: Examples of r-simple (left) and θ-simple (right) regions.
Math 250, Fall 2006 Revised December 6, 2006.