# How do you find the area of the region shared by the cardioid #r=2(1+cos(theta)) #and the circle r=2?

Let's find the points of intersection:

We can see from the graph that (considering the half of the area due to the symmetry and doubling the integrals):

By signing up, you agree to our Terms of Service and Privacy Policy

To find the area of the region shared by the cardioid (r=2(1+\cos(\theta))) and the circle (r=2), you first need to determine the points where the two curves intersect. These points will be the limits of integration. Then, you integrate the area between the curves using polar coordinates. The formula for finding the area between two curves in polar coordinates is (\frac{1}{2} \int_{\alpha}^{\beta} [f(\theta)]^2 - [g(\theta)]^2 d\theta), where (\alpha) and (\beta) are the angles at which the curves intersect, and (f(\theta)) and (g(\theta)) are the equations of the curves. After integrating, you will obtain the area of the region shared by the two curves.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you evaluate the definite integral #int (x^3-x^2)dx# from [1,3]?
- How do you find the antiderivative of #(e^x)/(1+e^(2x))#?
- How do you find the indefinite integral of #int (lnx)^2/x#?
- Let R be the region in the first quadrant bounded above by the graph of #y=(6x+4)^(1/2)# the line #y=2x# and the y axis, how do you find the area of region R?
- How do you find the integral of #sin^2(x)cos^6(x) dx#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7