Question

Sketch each region and use a double integral to find its area. The region inside both the cardioid \(r=1+\sin \theta\) and the cardioid \(r=1+\cos \theta\)

Short answer

Answer: The area of the region enclosed by the cardioids is \(3\pi - \frac{5\pi\sqrt{2}}{2} + \sqrt{2}\).

Step-by-step solution

Find the points of intersection between the cardioids

To find the points of intersection, we can set the two polar equations equal to each other and solve for the possible values of \(\theta\). \((1 + \sin \theta) = (1 + \cos \theta) \Rightarrow \sin \theta = \cos \theta\) From the last equation, we can find the values for \(\theta\) at the points of intersection: \(\theta = \frac{\pi}{4}\) and \(\theta = \frac{5\pi}{4}\)

Set up the double integral

In order to find the area enclosed by these curves, we must use a double integral and the Jacobian \(r\) for polar coordinates. Note that the limits of integration for \(\theta\) are determined by the points of intersection we found earlier. The double integral setup will look like this: \(\text{Area} = \int_{\frac{\pi}{4}}^{\frac{5\pi}{4}} \int_{1+\cos \theta}^{1+\sin \theta} r\ dr\ d\theta\)

Solve the double integral

Next, we will solve the double integral step by step. First, let's compute the inner integral: \(\int_{1+\cos \theta}^{1+\sin \theta} r\ dr = \left[\frac{1}{2}r^2\right]_{1+\cos \theta}^{1+\sin \theta} = \frac{1}{2}\left((1+\sin \theta)^2 - (1+\cos \theta)^2\right)\) Now, we need to compute the outer integral: \(\text{Area} = \int_{\frac{\pi}{4}}^{\frac{5\pi}{4}} \frac{1}{2}\left((1+\sin \theta)^2 - (1+\cos \theta)^2\right) d\theta\) To make it simpler, we can expand the terms inside the integral: \(\text{Area} = \frac{1}{2} \int_{\frac{\pi}{4}}^{\frac{5\pi}{4}} (\sin^2 \theta - \cos^2 \theta + 2\sin\theta - 2\cos\theta) d\theta\) Now, we can integrate term by term: \(\text{Area} = \frac{1}{2} \left[\int_{\frac{\pi}{4}}^{\frac{5\pi}{4}} \sin^2 \theta \ d\theta - \int_{\frac{\pi}{4}}^{\frac{5\pi}{4}} \cos^2 \theta \ d\theta + 2\int_{\frac{\pi}{4}}^{\frac{5\pi}{4}} \sin\theta \ d\theta - 2\int_{\frac{\pi}{4}}^{\frac{5\pi}{4}} \cos\theta \ d\theta \right]\) We can use the following trigonometric identity to simplify the integrals for \(\sin^2\theta\) and \(\cos^2\theta\): \(\sin^2 \theta = \frac{1-\cos(2\theta)}{2}, \ \cos^2\theta = \frac{1+\cos(2\theta)}{2}\) Then, our integrals become: \(\text{Area} = \frac{1}{2} \left[\int_{\frac{\pi}{4}}^{\frac{5\pi}{4}} \frac{1-\cos(2\theta)}{2} \ d\theta - \int_{\frac{\pi}{4}}^{\frac{5\pi}{4}} \frac{1+\cos(2\theta)}{2} \ d\theta + 2\int_{\frac{\pi}{4}}^{\frac{5\pi}{4}} \sin\theta \ d\theta - 2\int_{\frac{\pi}{4}}^{\frac{5\pi}{4}} \cos\theta \ d\theta \right]\) Finally, we can evaluate the integrals and simplify: \(\text{Area} = \frac{1}{2} \left[\frac{1}{2}\left(\frac{5\pi}{4} - \frac{\pi}{4}\right) - \frac{1}{4}\left(\sin(2\cdot\frac{5\pi}{4}) - \sin(2\cdot \frac{\pi}{4})\right) - \frac{5\pi}{4}\left(\cos\frac{5\pi}{4} - \cos\frac{\pi}{4}\right) - 2\left(\sin\frac{5\pi}{4} - \sin\frac{\pi}{4}\right) + 2\left(\cos\frac{5\pi}{4} - \cos\frac{\pi}{4}\right)\right]\) After simplifying, we get: \(\text{Area} = \frac{5\pi}{4} - \frac{\pi}{4} - \frac{1}{4}\left(-\sqrt{2}-\sqrt{2}\right) - \frac{5\pi}{4}\left(\sqrt{2} + \sqrt{2}\right)\) \(\text{Area} = 3\pi - \frac{5\pi\sqrt{2}}{2} + \sqrt{2}\) Thus, the area of the region enclosed by the cardioids \(r = 1 + \sin \theta\) and \(r = 1 + \cos \theta\) is \(3\pi - \frac{5\pi\sqrt{2}}{2} + \sqrt{2}\).

Key concepts

Polar Coordinates

Polar coordinates offer a unique way to describe the location of a point in a plane. Instead of traditional Cartesian coordinates, which use an \(x, y\) format to specify a point, polar coordinates determine a point's location based on a distance from a reference point and an angle from a reference direction.
The system is particularly useful in scenarios involving rotational symmetry or circular objects due to its simplicity in handling such shapes. A polar coordinate is given in the form of \(r, \theta\), where \(r\) is the radius or the distance from the origin, and \(\theta\) is the angle in radians from the positive x-axis.
For example, the equation of a circle centered at the origin is simply \(r = a\), where \(a\) is the radius of the circle. This simplicity often makes transforming problems into polar coordinates beneficial for finding areas, particularly when dealing with rotations or circular regions like the cardioids in this exercise.
When calculating areas using double integrals in polar coordinates, one must remember to include the Jacobian transformation, which in polar coordinates is just \(r\) itself. This adjustment is crucial to account for the transformation from Cartesian to polar planes, ensuring all area calculations remain accurate.

Cardioid

A cardioid is a heart-shaped curve commonly encountered in polar coordinates. It is expressed in an equation like \(r = 1 + \sin \theta\) or \(r = 1 + \cos \theta\).
The cardioid gets its name from its distinctive shape, resembling a heart. This shape arises from tracing a fixed point on a circle's circumference as it rolls around another circle with the same radius.
Cardioids have interesting properties that make them especially significant in mathematics and physics.

• They are examples of epicycloids, curves generated by a circle rolling around a circle.
• They have a single cusp, which is the pointed, heart-like indentation.
• They are symmetric relative to the horizontal axis in the case of \(r = 1 + \cos \theta\), and symmetric about the vertical axis for \(r = 1 + \sin \theta\).

In this exercise, we analyze the region enclosed by two intersecting cardioids. Calculating the intersections like this provides valuable insight into the shapes formed by curve plots and the areas these shapes enclose.

Area of Regions

Finding the area of regions bounded by curves, particularly in polar coordinates, offers a systematic approach to solving real-world and theoretical problems.
In our exercise, the focus is on the area enclosed by intersecting cardioids. The region's boundary occurs between the points of intersection of the two curves, \((1 + \sin \theta) = (1 + \cos \theta)\), solved to find angles \(\theta = \frac{\pi}{4}\) and \(\theta = \frac{5\pi}{4}\). These angles serve as the limits of integration.
To determine the area of such regions, a double integral in polar coordinates is employed:
\[\int_{\text{lower angle}}^{\text{upper angle}} \int_{\text{inner radius}}^{\text{outer radius}} r\ dr\ d\theta\]Here, \(r\) is the radial function defining the inner and outer edges of the region with respect to \(\theta \). Ensure always to set the correct boundaries based on the curves defining the region.
By resolving these integrals, you obtain the desired area. Such calculations make use of particular trigonometric identities and integration techniques, unraveling solutions to complex bounded regions in both academic and practical contexts.