Question on Area between Polar Curves

Hey everyone,

I have two questions regarding the area of polar curves.

1. Find the area of the region lying the polar curve r=1 + cos(theta), and outside the polar curve r= 2cos(theta)

2. Find the area of the shaded region inside the graph of r= 1+2cos(theta); (the graph shows the top half of the cardioid shaded)

Basically, I know how to solve these problem and how to draw the graphs. What I need help on is finding the limits of integration. For the 1st problem, I picked my limits of integration to be 0 to pi, and then I multiplied the integral by 2 to find the total area. (Is this method correct)

For the 2nd, I solved for r, and got cos(theta)= -1/2. So would my limits of integration be 2pi/3 to 4pi/3?

Any help and feedback appreciated.

Thanks

Re: Question on Area between Polar Curves

Quote:

Originally Posted by

**Beevo** Hey everyone,

I have two questions regarding the area of polar curves.

1. Find the area of the region lying the polar curve r=1 + cos(theta), and outside the polar curve r= 2cos(theta)

2. Find the area of the shaded region inside the graph of r= 1+2cos(theta); (the graph shows the top half of the cardioid shaded)

Basically, I know how to solve these problem and how to draw the graphs. What I need help on is finding the limits of integration. For the 1st problem, I picked my limits of integration to be 0 to pi, and then I multiplied the integral by 2 to find the total area. (Is this method correct)

For the 2nd, I solved for r, and got cos(theta)= -1/2. So would my limits of integration be 2pi/3 to 4pi/3?

Any help and feedback appreciated.

Thanks

In the first one, when you set up your double integral, for the top half, the radii are bounded above by $\displaystyle \displaystyle \begin{align*} r = 1 + \cos{\theta} \end{align*}$ and the radii are bounded below by $\displaystyle \displaystyle \begin{align*} r = 2\cos{\theta} \end{align*}$. I agree with your bounds for $\displaystyle \displaystyle \begin{align*} \theta \end{align*}$. So to find the area, your double integral is $\displaystyle \displaystyle \begin{align*} A = 2\int_0^{\pi}{\int_{2\cos{\theta}}^{1 + \cos{\theta}}{r\,dr}\,d\theta} \end{align*}$.

For part 2, you follow a similar process with the same $\displaystyle \displaystyle \begin{align*} \theta \end{align*}$ limits, and your radii are bounded above by $\displaystyle \displaystyle \begin{align*} r = 1 + 2\cos{\theta} \end{align*}$ and below by $\displaystyle \displaystyle \begin{align*} r = 0 \end{align*}$, giving your double integral for the area as $\displaystyle \displaystyle \begin{align*} A = \int_0^{\pi}{ \int_0^{1 + 2\cos{\theta}}{r\,dr} \,d\theta} \end{align*}$.

Re: Question on Area between Polar Curves

That makes sense, appreciate your help, thanks.

Re: Question on Area between Polar Curves

Hello, Beevo!

Quote:

1. Find the area of the region lying the polar curve $\displaystyle r\:=\:1 + \cos\theta$

and outside the polar curve $\displaystyle r\:=\: 2\cos\theta$

Basically, I know how to solve these problem and how to draw the graphs.

What I need help on is finding the limits of integration.

For the 1st problem, I picked my limits of integration to be 0 to pi,

and then I multiplied the integral by 2 to find the total area.

Is this method correct?

Yes . . . good work!

Quote:

2. Find the area of the shaded region inside the graph of $\displaystyle r \:=\:1 + 2\cos\theta$

(The graph shows the top half of the cardioid shaded.)

For the 2nd, I solved for r, and got cos(theta)= -1/2.

So would my limits of integration be 2pi/3 to 4pi/3? . . No

You solved $\displaystyle r \,=\,0$

You found the angles at which the curve passes through the pole (origin).

These are *not* the limits of integration. .(Well, probably not.)

There is yet another problem.

This is *not* a cardioid.

It is a *limacon* with an internal "loop".

So what region is shaded?

. . The entire upper half?

. . The upper half minus the loop?

. . Just the loop?

Re: Question on Area between Polar Curves

Quote:

Originally Posted by

**Soroban** Hello, Beevo!

Yes . . . good work!

You solved $\displaystyle r \,=\,0$

You found the angles at which the curve passes through the pole (origin).

These are *not* the limits of integration. .(Well, probably not.)

There is yet another problem.

This is *not* a cardioid.

It is a *limacon* with an internal "loop".

So what region is shaded?

. . The entire upper half?

. . The upper half minus the loop?

. . Just the loop?

Hey Soroban,

The entire upper half is shaded, including the loop. Would this change my limits of integration?

Thanks for your input