# Finding Area Between Two Polar Curves

• Lancelot59
In summary, the conversation discusses a problem involving finding the area inside two equations, r=sqrt(3)cos(theta) and r=sin(theta), which forms a petal shape. The solution involves using integration and the final answer is 17pi/4 + 3sqrt(3)/8. However, the book's answer is 5pi/24 - sqrt(3)/4, leading to a discussion about a possible typo in the book.
Lancelot59
This particular problem is just confusing me in the setup. I need to find the area that is inside both:
r=sqrt(3)cos(theta) and r=sin(theta)

It makes a petal type shape. I was beating my head around for a while, but I reasoned that since the equation used to find the area cuts out in a straight line. I could just move in either direction following the appropriate functions and get the area by adding the two parts together:

$$\frac{1}{2}[\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} (\sqrt{3}cos(\theta))^{2}) d\theta + \int_{0}^{\frac{\pi}{3}} (sin(\theta)^{2} d\theta]$$

It makes sense to me, but my final answer was 17pi\4 + 3sqrt(3)/8, however the book states the answer is 5pi/24 - sqrt3/4. Is my setup wrong, or did I just mess up with the integration?

Last edited:
You just messed up the integration.

It's probably just a typo, but the book's answer should be $5\pi/24-\sqrt{3}/4$.

It was a typo. The book had that answer. Thanks, I'll check my work again.

## 1. What is the formula for finding the area between two polar curves?

The formula for finding the area between two polar curves is ∫ab ½[r₁(θ)² - r₂(θ)²] dθ, where r₁(θ) and r₂(θ) are the equations of the two polar curves and a and b represent the angles at which the curves intersect.

## 2. How do I determine the limits of integration for finding the area between two polar curves?

The limits of integration can be determined by finding the points of intersection between the two polar curves. These points will represent the angles a and b in the formula for finding the area.

## 3. Can I use the same formula for finding the area between two polar curves if one curve is inside the other?

No, if one curve is entirely inside the other, the formula for finding the area between two polar curves will not work. In this case, you will need to find the area of the larger curve and subtract the area of the smaller curve.

## 4. Is there a graphical method for finding the area between two polar curves?

Yes, you can use a graphing calculator or software to plot the two polar curves and find the points of intersection. Then, you can use the "shaded region" function to find the area between the curves.

## 5. Can I use polar coordinates to find the area between two Cartesian curves?

No, the formula for finding the area between two polar curves is specifically for polar coordinates. If you want to find the area between two Cartesian curves, you will need to use the formula for finding the area between two functions in rectangular coordinates.

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