Chapter 27

The Integral in Other Coordinates27.1 Polar Coordinates

Recall the relation between the rectangular coordinates and polar coordinates is

x(r,θ)≡(

xy

)=

(r cos(θ)r sin(θ)

), r ≥ 0, θ ∈ [0,2π)

Now consider the part of grid obtained by fixing θ at various values and varying r and thenby fixing r at various values and varying θ .

The idea is that these lines obtained by fixing one or the other coordinate are veryclose together, much closer than drawn and so we would expect the area of one of thelittle curvy quadrilaterals to be close to the area of the parallelogram shown. Considerthis parallelogram. The two sides originating at the intersection of two of the grid lines asshown are approximately equal to

xr (r,θ)dr, xθ (r,θ)dθ

where dr and dθ are the respective small changes in the variables r and θ . Thus the areaof one of those little curvy shapes should be approximately equal to

|xr (r,θ)dr×xθ (r,θ)dθ |

by the geometric description of the cross product. These vectors are extended as 0 in thethird component in order to take the cross product. This reduces to

dA =

∣∣∣∣det(

cos(θ) −r sin(θ)sin(θ) r cos(θ)

)∣∣∣∣drdθ = rdrdθ

which is the increment of area in polar coordinates, taking the place of dxdy. The integralis really about taking the value of the function integrated multiplied by dA and adding theseproducts. Here is an example.

Example 27.1.1 Find the area of a circle of radius a.

The variable r goes from 0 to a and the angle θ goes from 0 to 2π . Therefore, the areais ∫

DdA =

∫ 2π

0

∫ a

0rdrdθ = πa2

Example 27.1.2 The density equals r. Find the total mass of a disk of radius a.

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