Change of Order of Integration
(Page: 1
 2 ) Text If we apply Fubini's Theorem to integrals using polar coordinates, we get
∫_{a}^{b}∫_{c}^{d}f(x, y)rdrdθ = ∫_{c}^{d}∫_{a}^{b}f(x, y)rdθdr = ∫∫_{R}f(r, θ)dA
Where dA = dr*dθ.
Demos
Change of Order of Integration
 
In this demo, start with "step" and "step2" set to their minimum values. Increase "step" to its maximum first and then increase "step2" to its maximum to represent integration with respect to r first and θ second. To see the alternative order of integration, increase "step2" first.

Exercises One of the change of order of integration demos for Cartesian coordinates discusses "slab approximations". What would slabs look like in polar coordinates? How would you use summation and integral notation to describe slab approximations in polar coordinates?
