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Chapter 3: Problem 4

Find the area common to the cardiod \(r=a(1+\cos \theta)\) and the circle \(\mathrm{r}=\frac{3}{2} \mathrm{a}\), and also the area of the remainder of the cardiod.

Short Answer

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Question: Find the area of the remainder of the cardioid after subtracting the area common to both the given cardioid and circle. Given: Cardioid: \(r = a(1 + \cos \theta)\) Circle: \(r = \frac{3}{2}a\) Solution: 1. Identify the given equations. 2. Find the points of intersection between the cardioid and the circle. 3. Calculate the area common to both curves. 4. Find the total area of the cardioid. 5. Subtract the common area from the total cardioid area to find the area of the remainder of the cardioid. Answer: Follow the steps in the solution to find the area of the remainder of the cardioid.

Step by step solution

01

Identify the given equations

We have been given: Cardioid: \(r = a(1 + \cos \theta)\) Circle: \(r = \frac{3}{2}a\)
02

Find the points of intersection

To find the intersections, we need to solve the equation: \(a(1 + \cos \theta) = \frac{3}{2}a\) Divide both sides by \(a\) to get: \(1 + \cos \theta = \frac{3}{2}\) Now, solve for \(\cos \theta\): \(\cos \theta = \frac{1}{2}\) Because the cosine function is positive in the first and fourth quadrants, we have two intersection points corresponding to angles \(\theta_{1} = \frac{\pi}{3}\) and \(\theta_{2} = \frac{5\pi}{3}\).
03

Find the area common to both curves

To find the area common to both curves, we'll integrate both functions from \(\theta_{1}\) to \(\theta_{2}\) and subtract the smaller area from the larger area. The integral representation for polar coordinates area is \(\frac{1}{2} \int_{\theta_1}^{\theta_2} r^2 \,d\theta\). Integrating both functions from \(\theta_{1}\) to \(\theta_{2}\), we get: \(A_\text{cardioid} = \frac{1}{2} \int_{\frac{\pi}{3}}^{\frac{5\pi}{3}} a^2(1+\cos \theta)^2 \,d\theta\) \(A_\text{circle} = \frac{1}{2} \int_{\frac{\pi}{3}}^{\frac{5\pi}{3}} \left(\frac{3}{2}a\right)^2 \,d\theta\) Calculate \(A_\text{common} = A_\text{cardioid} - A_\text{circle}\).
04

Find the total area of the cardioid

To find the total area of the cardioid, we'll integrate its equation from \(0\) to \(2\pi\): \(A_\text{total} = \frac{1}{2} \int_{0}^{2\pi} a^2(1+\cos \theta)^2 \,d\theta\)
05

Find the area of the remainder of the cardioid

To find the remaining area of the cardioid, we simply subtract the common area from the total cardioid area: \(A_\text{remainder} = A_\text{total} - A_\text{common}\) By calculating \(A_\text{remainder}\), we have found the area of the remainder of the cardioid.

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