Find the area that is common to the cardioids with equations

r = 1 + cos θ and r = 1 - cos θ.

Type your answer in the space below and give 3 decimal places. If your answer is less than 1, place a leading "0" before the decimal point (ex: 0.482). (10 points)

If you plot the two cardioids, you see that they intersect at θ = π/2, 3π/2

Using symmetry of the curves, you see that the area is
a = 4∫[π/2,π] 1/2 (1+cosθ)^2 dθ = 3π/2 - 4

To find the area that is common to the cardioids with equations r = 1 + cos(θ) and r = 1 - cos(θ), we need to determine the limits of integration for θ and integrate the area between the curves.

First, let's solve the equations to find the intersection points:

1 + cos(θ) = 1 - cos(θ)
2cos(θ) = 0
cos(θ) = 0

The solutions for cos(θ) = 0 are θ = π/2 and θ = 3π/2.

Now, let's integrate the difference in the areas between the curves from θ = π/2 to θ = 3π/2.

The equation for the area between two polar curves is given by:

A = (1/2) ∫ [r_outer(θ)^2 - r_inner(θ)^2] dθ,

where r_outer(θ) is the equation for the curve outside and r_inner(θ) is the equation for the curve inside.

In this case, the outer curve is r = 1 + cos(θ) and the inner curve is r = 1 - cos(θ).

Using the equation for the area:

A = (1/2) ∫ [(1 + cos(θ))^2 - (1 - cos(θ))^2] dθ

Simplifying the equation inside the integral:

A = (1/2) ∫ [1 + 2cos(θ) + cos^2(θ) - 1 + 2cos(θ) - cos^2(θ)] dθ
A = (1/2) ∫ [4cos(θ)] dθ
A = 2 ∫ cos(θ) dθ
A = 2 sin(θ)

Integrating 2 sin(θ) from θ = π/2 to θ = 3π/2:

A = 2[sin(3π/2) - sin(π/2)]
A = 2[-1 - 1] # since sin(3π/2) = -1 and sin(π/2) = 1
A = 2(-2)
A = -4

Since the value of -4 is less than 1, we place a leading "0" before the decimal point. Therefore, the area that is common to the cardioids is 0.000.

To find the area that is common to the two cardioids, we need to determine the bounds of integration for both the radius (r) and the angle (θ).

First, let's find the bounds of integration for r by setting the two equations equal to each other:
1 + cos θ = 1 - cos θ.

By simplifying the equation, we get:
2cos θ = 0.

Dividing both sides by 2, we obtain:
cos θ = 0.

This equation is true when θ = π/2 and θ = 3π/2. So the bounds of integration for r are from 0 to π/2 and from π to 3π/2.

Next, let's find the bounds of integration for θ by setting the two equations equal to each other:
1 + cos θ = 1 - cos θ.

Canceling out the common terms, we get:
2cos θ = 0.

This equation is true when θ = π/2 and θ = 3π/2. So the bounds of integration for θ are from π/2 to 3π/2.

Now, we can set up the double integral to find the area:

A = ∬R r dr dθ,

where R represents the common region between the two cardioids.

The area can be evaluated using a double integral with the given bounds:

A = ∫(π/2 to 3π/2) ∫(0 to 1 + cos θ) r dr dθ.

Evaluating this integral will yield the desired area, which is the area common to the two cardioids.