Subsections

• Area of Polar Graphs

Maple Commands

>plot(cos(2*theta),theta=0..2*Pi,coords=polar);

Don't forget the option ,coords=polar! This graph is a four-leafed rose. Polar graphs can be hard to understand. Animating the graph as the angle increases will help.

>with(CalcP7):
>ParamPlot([cos(2*t),t],t=0..2*Pi,coords=polar);

When you run the ParamPlot command, you first get a set of axes with no curves drawn, and you think that there is something wrong. What you need to do to see the curves is first click on the graph. A box appears around the graph and a set of controls appears in the context bar just below the menu shortcut buttons at the top of the main Maple window. The set of controls works like those on a VCR. To see the animated graph, click on the play button. The other controls in the context bar allow you to slow down or speed up the animation, step through the animation one frame at a time, stop the animation, and even run the animation in reverse. I suggest you play with them until you feel comfortable. To find where two graphs intersect you set the functions equal to each other as they both equal the radius and then solve for the angle.

>plot({2-sin(theta),3*sin(3*theta)},theta=0..Pi,coords=Polar);

As discussed above there can be infinite solutions so use the fsolve command and choose a range of angles values in which the intersection point occurs.

>a:=fsolve(2-sin(theta)=3*sin(3*theta),theta=0..0.5);
>b:=fsolve(2-sin(theta)=3*sin(3*theta),theta=0.5..1.5);
>c:=fsolve(2-sin(theta)=3*sin(3*theta),theta=1.5..2.5);
>d:=fsolve(2-sin(theta)=3*sin(3*theta),theta=2.5..3.5);

The following commands find the radius vaue for each angle.

>2-sin(a);
>2-sin(b);
>2-sin(c);
>2-sin(d);

Area of Polar Graphs

The formula for area inside $r=f(\theta)$ in polar coordinates is given by

$$\frac{1}{2}\int_{\theta_1}^{\theta_2}r^2 d\theta = \frac{1}{2}\int_{\theta_1}^{\theta_2}f(\theta)^2 d\theta$$

. This comes from the fact that the area in a thin wedge with radius $r$ and angle $\Delta \theta$ is $\frac{1}{2}r^2\Delta\theta$. The command below shows how to find the area of a cardioid in Maple.

>evalf(1/2*int((1-\cos(theta))^2,theta=0..2*Pi));