The area shared by two polar curves

[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?

 

[Mark44]

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.

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.

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.