Knowing the equation of the circle (x -20)^2 + y^2 = 225 in rectangular coordinates and the equation of the ellipse r = 9/(1-0.8cos(theta)) in polar coordinates, find the shaded area. (Hint: The circle and the ellipse are concentric)
The shaded area is outside the circle but inside the ellipse.
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To find the shaded area, we need to compute the area of the region between the circle and the ellipse.
First, let's simplify the equation of the ellipse in polar coordinates:
r = 9 / (1 - 0.8cos(theta))
Multiply both sides by (1 - 0.8cos(theta)):
r(1 - 0.8cos(theta)) = 9
Expand the left side of the equation:
r - 0.8cos(theta)r = 9
r = 9 + 0.8cos(theta)r
Now, let's convert the equation of the circle to polar coordinates:
(x - 20)^2 + y^2 = 225
r^2 - 40rcos(theta) + 400 + r^2sin^2(theta) = 225
2r^2 - 40rcos(theta) + 400 = 225
2r^2 - 40rcos(theta) + 175 = 0
Since the circle and the ellipse are concentric, their radii are equal at the points of intersection.
Therefore, we can set the two equations equal to each other:
9 + 0.8cos(theta)r = 2r^2 - 40rcos(theta) + 175
Rearrange the equation:
2r^2 - 40rcos(theta) -0.8cos(theta)r + 175 - 9 = 0
2r^2 - 40rcos(theta) - 0.8cos(theta)r + 166 = 0
Now, we can find the values of r where the circle and the ellipse intersect by solving this equation.
To find the area between the circle and the ellipse, we integrate the difference in their areas over the range of theta where they intersect.
Let's denote the region of intersection as R.
The area of the circle is given by A1 = ∫[R](0.5r^2)d(theta)
The area of the ellipse is given by A2 = ∫[R](0.5r^2) d(theta)
Subtracting the two equations, we get:
A = A2 - A1 = ∫[R] (0.5r^2 - 0.5r^2) d(theta)
A = ∫[R] 0 d(theta)
Since the integrand is 0, the integral evaluates to 0.
Therefore, the shaded area between the circle and the ellipse is 0.
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