Chapter 7: Additional Applications of Integration
Section 7.2: Integration in Polar Coordinates
Table 7.2.1 lists integration formulas for the curve r=rθ given in polar coordinates.
The arc length integral is derived from the parametric form given at the end of Table 5.4.1.
(See Example 7.2.9.)
The area integral is derived from A=r2θ/2, the expression for the area of a sector of a circle.
(See Example 7.2.10.)
Table 7.2.1 Integration in polar coordinates
The shaded region in Figure 7.2.1 is a geometric representation of the element of area in polar coordinates. The region is bounded by the two green radii and some polar curve r=rθ.
The angles corresponding to the two bounding radii are θ and θ+dθ, so that the vertex angle of the sector is dθ.
The area of the sector is approximately that of a triangle with altitude r and base s=r dθ. Hence, the area of this "triangle" is r s/2=r2dθ/2.
The area between an outer polar curve r=fθ and an inner curve r=gθ can be found by the subtraction
12∫θ1θ2f2 ⅆθ−12∫θ1θ2g2 ⅆθ=12∫θ1θ2f2−g2 ⅆθ
Figure 7.2.1 Element of area in polar coordinates
provided the angles θ1 and θ2 are the same for each curve.
Working in polar coordinates, calculate the area of the circle r=2 a cosθ.
Working in polar coordinates, calculate the circumference of the circle r=2 a cosθ.
Working in polar coordinates, calculate the area enclosed by the cardioid r=1+ cosθ.
Working in polar coordinates, calculate the arc length of cardioid r=1+cosθ.
Working in polar coordinates, calculate the complete arc length of the limaçon r=1/2−cosθ.
Working in polar coordinates, calculate the area within the inner loop of the limaçon r=1/2−cosθ.
Working in polar coordinates, calculate the area outside the circle r1=2, but inside the cardioid r2=21+cosθ.
Working in polar coordinates, calculate the area common to the circle r1=sinθ and the inner loop of the limaçon r2=1/5+cosθ.
Starting with the expression for the arc length of a curve defined parametrically, obtain the expression for the arc length of a curve defined by r=rθ in polar coordinates. (See Table 7.2.1.)
Obtain dA=12r2dθ, the area element in polar coordinates.
<< Previous Section Table of Contents Next Section >>
© Maplesoft, a division of Waterloo Maple Inc., 2023. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document