Section 4.4 Calculus 2 - Professor Clontz

4.4 Areas and Lengths using Polar Coordinates

The area of the circle sector of angle \(\theta\) is given by \(A=\pi r^2\times\frac{\theta}{2\pi}=\frac{1}{2}r^2\theta\).
Therefore the area bounded by \(\alpha\leq\theta\leq\beta\) and \(r=r(\theta)\) is \(A=\frac{1}{2}\int_\alpha^\beta(r(\theta))^2\,d\theta\).
Example Find the area bounded by the cardioid \(r=2+2\sin\theta\).
To obtain the area where \(r(\theta)\leq r\leq R(\theta)\), where \(r\) is an inside curve and \(R\) is an outside curve, find the clockwise angle \(\alpha\) and counter-clockwise angle \(\beta\) where they intersect, and use \(A=\frac{1}{2}\int_\alpha^\beta((R(\theta))^2-(r(\theta))^2)\,d\theta\).
Example Find the area outside the circle \(x^2+y^2=1\) and inside the cardioid \(r=1-\cos\theta\).

The polar curve \(r(\theta)\) where \(\alpha\leq\theta\leq\beta\) may be parametrized by \(x=r(\theta)\cos\theta,y=r(\theta)\sin\theta\).
Therefore its length is given by \(L=\int_\alpha^\beta\sqrt{ (\frac{dx}{d\theta})^2+(\frac{dy}{d\theta})^2 }\,d\theta\) which simplifies to \(L=\int_\alpha^\beta\sqrt{ (r(\theta))^2+(r’(\theta))^2 }\,d\theta\).
Example Show that the circumference of the circle of radius \(R\) is \(2\pi R\).
Example Find the circumference of the spiral \(r=\theta^2\) from \(p(0,0)\) to \(p(5,\sqrt 5)\).

Find the area inside \(r=\cos2\theta\) where \(0\leq\theta\leq\pi/4\).
Find the area bounded by the cardioid \(r=1-\cos\theta\).
Sketch the region where \(|x|\leq y\leq\sqrt{1-x^2}+1\). Show that its area is \(\frac{\pi}{2}+1\). (Hint: Show that this is the area inside \(r=2\sin\theta\) where \(\pi/4\leq\theta\leq3\pi/4\).)
Find the length of the first rotation of the spiral \(r=e^\theta\).
Use the polar arclength formula to show that the circumference of the circle \(r=4\sin\theta\) is \(4\pi\).
Show that the length of the cardioid \(r=2+2\cos\theta\) is \(\int_0^{2\pi}\sqrt{8+8\cos\theta}\,d\theta=16\).
Prove that if \(x=r(\theta)\cos\theta\) and \(y=r(\theta)\sin\theta\) then \(\int_\alpha^\beta\sqrt{ (\frac{dx}{d\theta})^2+(\frac{dy}{d\theta})^2 }\,d\theta =\int_\alpha^\beta\sqrt{ (r(\theta))^2+(r’(\theta))^2 }\,d\theta\).
What definite integral is the area of the cardioid \(r=4+4\sin\theta\)?
What definite integral is the length of the curve \(r=\cos^2\theta\) where \(0\leq\theta\leq\pi/2\)?