Chapter 7: Additional Applications of Integration
Section 7.2: Integration in Polar Coordinates
Example 7.2.3
Working in polar coordinates, calculate the area enclosed by the cardioid r=1+ cosθ.
Solution
Mathematical Solution
Figure 7.2.3(a) animates the polar area-element as it sweeps through the cardioid r=1+cosθ. The calculation of the area is based on the expression given in Table 7.2.1.
12∫02 π1+cosθ2 ⅆθ
=12∫02 π1+2 cosθ+cos2θ ⅆθ
=12∫02 π1+2 cosθ+1+cos2 θ2 ⅆθ
=12∫02 π32 ⅆθ+∫02 π2 cosθ+cos2 θ2 ⅆθ
=1232 2 π+0+0
=3 π/2
use plots, plottools in module() local R,X,Y,P,G; P := plot(1+cos(theta), theta = 0 .. 2*Pi, coords = polar, scaling = constrained,color=black,tickmarks=[3,3]): R := 1+cos(t): X := unapply(R*cos(t), t): Y := unapply(R*sin(t), t): G:=proc(t) display(polygon([[0,0],[X(t),Y(t)],[X(t+.1),Y(t+.1)]],color=red)); end: print(animate(G,[theta],theta=0..2*Pi-.1,background=P,frames=62)); end module; end use:
Figure 7.2.3(a) Animation: area element for cardioid
The calculation is simplified by noting that the integrals of cosθ and cos2 θ over the interval 0,2 π are both zero.
Maple Solution
A stepwise solution similar to the one at the left of Figure 7.2.3(a) can be obtained with the tutor. Setting the Constant Multiple and Sum rules as Understood Rules helps eliminate some of the tedium of working through such a stepwise solution.
The area is most efficiently obtained interactively as follows.
Expression palette: Definite Integral template
Context Panel: Evaluate and Display Inline
12∫02 π1+cosθ2 ⅆθ = 32π
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