Find the area of the region lying inside the polar curve r = 2(1 − cos theta) and outside the polar curve r + 4 cos(theta)= 0.
To find the area of the region lying inside one polar curve and outside another polar curve, you need to integrate the difference of their respective areas between two values of theta.
The first step is to determine the points where the two curves intersect. Set the equations of the curves equal to each other and solve for theta:
2(1 − cos(theta)) = -4 cos(theta)
2 - 2cos(theta) = -4 cos(theta)
6 cos(theta) = 2
cos(theta) = 1/3
Using the identity cos(theta) = x/r, where x and r are the Cartesian coordinates, we can convert the coordinate of the intersection point from polar coordinates (r, theta) to Cartesian coordinates (x, y):
x = r * cos(theta) = (-4 cos(theta)) * cos(theta) = -4 cos^2(theta) = -4 (1 - sin^2(theta)) = -4 + 4 sin^2(theta)
y = r * sin(theta) = (-4 cos(theta)) * sin(theta) = -4 cos(theta) sin(theta) = (-2 sin(2 theta))
Therefore, the intersection point in Cartesian coordinates is (x, y) = (-4 + 4 sin^2(theta), -2 sin(2 theta)).
To calculate the area between the curves, you need to set up the integral. The area is given by the formula:
A = ∫[θ1,θ2] (1/2) [(r2)^2 - (r1)^2] dθ
Since r2 is the outer curve and r1 is the inner curve, the integral becomes:
A = ∫[θ1,θ2] (1/2) [(2(1 − cos(theta)))^2 - ((-4 cos(theta))^2)] dθ
Now, we need to find the limits of integration, θ1 and θ2, which correspond to the points of intersection of the two curves.
Since the curves intersect at θ where cos(theta) = 1/3, we can use the inverse cosine function to find the limits:
θ1 = arccos(1/3)
θ2 = -arccos(1/3)
Finally, you can evaluate the integral to find the area of the region.
A = ∫[arccos(1/3), -arccos(1/3)] (1/2) [(2(1 − cos(theta)))^2 - ((-4 cos(theta))^2)] dθ
To simplify the integral, you can expand and simplify the expressions in the integrand and then integrate each term separately.
After computing the integral, you will have the area of the region lying inside the polar curve r = 2(1 − cos(theta)) and outside the polar curve r + 4 cos(theta) = 0.