# Section 4.4 Calculus 2

Areas and Lengths using Polar Coordinates

## 4.4 Areas and Lengths using Polar Coordinates

### 4.4.1 Area Between Polar Curves

• 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 $$r=1-\cos\theta$$.

### 4.4.2 Length of a Polar Curve

• 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 $$p(5,\sqrt 5)$$.

### Review Exercises

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

Solutions 1-7

Solutions 8-9

### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 10.5