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.
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