# The area shared by two polar curves

• rdioface
In summary, to find the area of the lemniscate r2=6cos(2w) located to the right of the line r=3/2sec(w), you will need to solve the equation 24/9*cos(2w)=sec2(w) to find the points of intersection, and then set up an integral using the formula for the area of a polar region. You can take advantage of symmetry to simplify the calculation.
rdioface

## Homework Statement

Find the area of the lemniscate r2=6cos(2w) located to the right of the line r=3/2sec(w).

## Homework Equations

Area of a polar region is the integral from a to b (in this case a and b are where the curve intersects the line, I believe) of 1/2*r2d(w).

## The Attempt at a Solution

An attempt to find the points of intersection yield the equation 24/9*cos(2w)=sec2(w), which I have no idea how to solve. Additionally I'm not quite sure how to set up the integral in terms of what area minus what. I know it should be similar to the form for Cartesian int[a,b]f(x)dx-g(x)dx, but how am I to know which one is "lower" in terms of polar co-ordinates?

rdioface said:

## Homework Statement

Find the area of the lemniscate r2=6cos(2w) located to the right of the line r=3/2sec(w).
Your line equation is written correctly, but for clarity it would be better as r = (3/2)secw.
rdioface said:

## Homework Equations

Area of a polar region is the integral from a to b (in this case a and b are where the curve intersects the line, I believe) of 1/2*r2d(w).

## The Attempt at a Solution

An attempt to find the points of intersection yield the equation 24/9*cos(2w)=sec2(w), which I have no idea how to solve.
Replace cos(2w) by 2cos2(w) - 1, and replace sec2(w) by 1/cos2(w). When you multiply the resulting equation by cos2(w) you'll get a 4th degree equation that is quadratic in form.
rdioface said:
Additionally I'm not quite sure how to set up the integral in terms of what area minus what. I know it should be similar to the form for Cartesian int[a,b]f(x)dx-g(x)dx, but how am I to know which one is "lower" in terms of polar co-ordinates?

Have you sketched the region whose area you are to find? The graph of the lemniscate is a sort of figure 8 lying horizontally. The region whose area you want to find is in the right-hand branch and to the right of the vertical line x = 3/2.

To make life easier you can take advantage of the symmetry of the lemniscate and vertical line, and simply calculate the area of the upper portion and double your result.

In your integrand the "upper" curve (larger r value) is the lemniscate. The "lower" curve (smaller r value) is the line.

## 1. What is the area shared by two polar curves?

The area shared by two polar curves is the region that is enclosed by the two curves when they intersect. It is often represented as a shaded region on a polar graph.

## 2. How do you find the area shared by two polar curves?

To find the area shared by two polar curves, you can use the formula A = 1/2∫θ1θ2 (r1)2 - (r2)2 dθ, where θ1 and θ2 are the angles of intersection between the two curves, and r1 and r2 are the corresponding polar equations for each curve.

## 3. Can the area shared by two polar curves be negative?

No, the area shared by two polar curves cannot be negative. The formula for finding the area is based on the concept of finding the difference between two positive values, resulting in a positive area.

## 4. What if the two polar curves do not intersect?

If the two polar curves do not intersect, then there is no shared area between them. In this case, the area shared by two polar curves would be equal to 0.

## 5. Are there any special cases when finding the area shared by two polar curves?

Yes, there are a few special cases when finding the area shared by two polar curves. These include when one curve is entirely inside the other, when the two curves are identical, or when the curves have multiple intersections. In these cases, the formula for finding the area may need to be modified to accurately calculate the shared area.

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