Advanced Engineering Mathematics, 7th Edition

9.11 Double Integrals in Polar Coordinates

INTRODUCTION

A double integral, which may be difficult or even impossible to evaluate in rectangular xy-coordinates, may become more tractable when expressed in a different coordinate system. In this section we examine double integrals in polar rθ-coordinates.

Polar Rectangles

Suppose R is a region in the plane bounded by the graphs of the polar equations r = g₁(θ), r = g₂(θ), and the rays θ = α, θ = β, and f is a function of r and θ that is continuous on R. In order to define the double integral of f over R, we use rays and concentric circles to partition the region into a grid of polar rectangles ...