Find the area shared by the cardioids r=6(1+cos(theta)) and r=6(1-cos(theta))
The curves intersect when θ= π/2,3π/2
By symmetry, then the area is
a = 2∫[π/2,3π/2] 1/2 (6(1-cosθ))^2 dθ
= 36∫[π/2,3π/2] 1 - 2cosθ + cos^2θ dθ
= 9(6θ-8sinθ+sin2θ) [π/2,3π/2]
= 4 + 3π/2
To find the area shared by the cardioids, we need to calculate the area enclosed by each cardioid and subtract the area outside their intersection.
First, let's find the points of intersection between the two cardioids. To do this, we equate their radial equations:
6(1 + cos(theta)) = 6(1 - cos(theta))
Now, we simplify the equation:
1 + cos(theta) = 1 - cos(theta)
2cos(theta) = 0
cos(theta) = 0
At this point, we find the values of theta that satisfy this equation. Since cos(theta) = 0 when theta = π/2 + nπ, where n is an integer, we have two points of intersection: θ = π/2 and θ = 3π/2.
Now, let's calculate the areas of the cardioids separately. The area bounded by a polar curve r = f(theta) can be obtained by integrating 0.5*r^2 with respect to theta over the appropriate interval.
For the first cardioid r = 6(1 + cos(theta)), we calculate its area:
A1 = 0.5 * ∫[θ = 0 to π/2] (6(1 + cos(theta)))^2 dθ
Simplifying this expression, we get:
A1 = 0.5 * ∫[θ = 0 to π/2] 36(1 + 2cos(theta) + cos^2(theta)) dθ
Now, we integrate, keeping in mind that cos^2(theta) = 0.5(1 + cos(2theta)):
A1 = 0.5 * ∫[θ = 0 to π/2] 36(1 + 2cos(theta) + 0.5(1 + cos(2theta))) dθ
A1 = 0.5 * ∫[θ = 0 to π/2] (54 + 36cos(theta) + 18cos(2theta)) dθ
Evaluating this integral, we get the area A1.
Next, we calculate the area A2 enclosed by the second cardioid r = 6(1 - cos(theta)):
A2 = 0.5 * ∫[θ = π/2 to 3π/2] (6(1 - cos(theta)))^2 dθ
Using similar steps as before, we integrate and evaluate this integral to get the area A2.
Finally, we calculate the area shared by the two cardioids by subtracting the area outside their intersection:
Area_shared = A1 - A2
Evaluate this expression to find the area shared by the two cardioids.