Section 16.4Double Integrals In Polar

Coordinates

• Thus far when we have integrated over a region R, we have used rectangular coordinates

• There may be situations where it is easier to use polar coordinates to define R

• If a region R or curve C is symmetric with respect to the origin (or part of such a region/curve), then polar coordinates may be a simpler description of the region or curve

• Where rectangular coordinates require the horizontal and vertical distance from the origin, polar coordinates require the distance from the origin and the angle between the ray the point lies on and the positive side of the horizontal axis

• Suppose P is the indicated point in the xy-plane• Then

• What are the rectangular coordinates given by (5,π/6)?

• What are the polar coordinates given by (-4, 5)

. P(x,y) = P(r,θ)

x

y

r

θ

0,tan

and

sincos

22

xx

y

yxr

ryrx

Write the following regions in both rectangular and polar form

x

y

(5,0)

(0,5)

4 {2 {

Converting our Integral

• We need

• Now g(r,θ) = f(rcosθ, rsinθ)• What about dA?• Just as dA in rectangular coordinates is made up of

a rectangular grid, dA in polar coordinates is made up of a polar grid

R R

dArgdAyxf ),(),(

Recall area of a sector

x

y

θ

r

22

1rA

• Using this info let’s see how we can figure out dA

Computing Polar Integrals• So to compute polar integrals we have

• Giving us

222 ryx

sincos ryrx

drdrdAddrrdA or

R R

ddrrrgdAyxf ),(),(

Example• Find the volume of a cylinder of radius a and

height h• Find the volume under the plane z = x and

above the disk of radius 3 bounded by x ≥ 0 using both rectangular and polar coordinates

• Let’s take a look at an example with Maple