Question

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

Short answer

The area of the region bounded by the two cardioids is 2 square units.

Step-by-step solution

Sketch the cardioids

Start by plotting each of the cardioids separately. The cardioid \(r=1+\sin\theta\) has a minimum distance of 1 at \(\theta = \pi\) and a maximum distance of 2 at \(\theta = 0\). Similarly, the cardioid \(r = 1 + \cos\theta\) has a minimum distance of 1 at \(\theta = \frac{\pi}{2}\) and a maximum distance of 2 at \(\theta = \pi\). Now, the two cardioids intersect as they both have a maximum distance of 2 from the origin. You can find the intersecting points that we will call \(\theta_1\) and \(\theta_2\) by setting the two equations to equal each other.

Find the intersection points

To find the intersection points, set the two equations equal to each other and solve for \(\theta\): \(1+\sin\theta = 1+\cos\theta\) Subtract 1 from both sides and simplify: \(\sin\theta = \cos\theta\) Divide both sides by \(\cos\theta\): \(\tan\theta = 1\) Since \(\tan\theta = 1\) in the first and second quadrants, we have two intersection points. The first intersection point occurs in the first quadrant at \(\theta_1 = \frac{\pi}{4}\) and the second intersection point occurs in the second quadrant at \(\theta_2 = \frac{3\pi}{4}\).

Find the area

Now, we'll use the polar integral formula to find the area in between the two cardioids. The formula for area in polar coordinates is given by: $$\frac{1}{2} \int_{\theta_1}^{\theta_2} [r^2(\theta)] d\theta.$$ Remember that we want to find the area inside both cardioids: $$Area = \frac{1}{2} \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} [(1+\sin\theta)^2 - (1+\cos\theta)^2] d\theta.$$

Integrate and solve

Now. we'll integrate with respect to \(\theta\). $$Area = \frac{1}{2} \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} (1+2 \sin{\theta} + \sin^2{\theta} - 1 - 2\cos\theta - \cos^2\theta) d\theta$$ Using the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\), we can see that the integral becomes: $$Area = \frac{1}{2} \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} (2\sin\theta - 2\cos\theta) d\theta$$ Now integrate: $$Area = \frac{1}{2} [-2\cos\theta - 2\sin\theta] \bigg|_{\frac{\pi}{4}}^{\frac{3\pi}{4}}$$ Finally, plug in the values: $$Area = \frac{1}{2} [-2(\cos\frac{3\pi}{4}-\cos\frac{\pi}{4}) - 2(\sin\frac{3\pi}{4}-\sin\frac{\pi}{4})] = 2.$$ Thus, the area of the region bounded by the two cardioids is 2 square units.

Key concepts

Cardioid

A cardioid is a fascinating type of curve that resembles a heart shape, often described in polar coordinates. It's particularly interesting because it's formed by a combination of circular motions, resulting in this unique shape. The main equations for cardioids are given by the polar equations \(r = 1 + \sin \theta\) and \(r = 1 + \cos \theta\). These formulas emerge from adding the sine or cosine function to a circular motion with a constant radius. Each of these cardioids behaves differently:

• For the cardioid \(r = 1 + \sin \theta\), the curve has its farthest distance from the origin (maximum) at \(\theta = 0 \) and closest distance (minimum) at \(\theta = \pi \).
• The cardioid \(r = 1 + \cos \theta\) is furthest at \(\theta = \pi \) and nearest at \(\theta = \frac{\pi}{2}\) from the origin.

These cardioids intersect in a figure-eight like shape when plotted together. Understanding where and how these curves intersect is key in further efforts like calculating enclosed areas or transformations.

Integration

Integration in polar coordinates can be a bit different from what you might be used to with Cartesian coordinates. Yet, it's a really effective method when dealing with circular or radial shapes like cardioids. In this context, integration allows us to calculate areas within these curves by summing up infinitely small wedges.The formula for finding the area of a region in polar coordinates is:\[ \text{Area} = \frac{1}{2} \int_{\theta_1}^{\theta_2} [r^2(\theta)] d\theta \]By integrating between two angles \(\theta_1\) and \(\theta_2\), which represent the intersection points of the cardioids, the above formula computes the area of the enclosed region. It's crucial to note that this formula integrates the square of the radius because the area swept by the angle \(d\theta\) forms a sector of a circle, represented mathematically by \(r^2(\theta)\).When solving our specific example, we substitute the equations for each cardioid into the integration formula to find the area between the two curves. Thus, mastering integration in polar coordinates becomes invaluable for finding such areas efficiently.

Area Calculation

Area calculation in case of our intersecting cardioids is carried out by determining the overlap between the two curves through polar integration. Here's how you can compute it step-by-step:1. **Intersecting Angles:** The first step is to identify angles where cardioids intersect. These are solved by equating the equations \(1 + \sin \theta = 1 + \cos \theta\). Solving this gives two intersection points at \(\theta = \frac{\pi}{4}\) and \(\theta = \frac{3\pi}{4}\), demarcating where you need to integrate between. 2. **Polar Integral:** We use the integral \[ \frac{1}{2} \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} ((1+\sin\theta)^2 - (1+\cos\theta)^2) d\theta \] to calculate the area. This integral subtracts the area under the \(\cos\) cardioid from the \(\sin\) cardioid, reflecting the actual intersecting shape. 3. **Integration Process:** Finally, computing this integral necessitates expanding the squared terms, simplifying using trigonometric identities, and integrating the resulting expression from \(\theta = \frac{\pi}{4}\) to \(\theta = \frac{3\pi}{4}\). 4. **Result:** As a result, the integration yields an area of 2 square units between the two cardioids. This step-by-step process underscores the importance of precise calculation and comprehension of both integration and polar concepts.