Problem 46 - Calculus Early Transcendentals

Question

Find the following average values. The average distance between points within the cardioid \(r=1+\cos \theta\) and the origin

Step-by-Step Solution

Answer

Answer: The average distance between points within the cardioid \(r=1+\cos \theta\) and the origin is 1.

Step 1: Convert to Cartesian Coordinates

Convert the cardioid equation \(r=1+\cos \theta\) to Cartesian coordinates \((x,y)\) using polar-to-cartesian transformations: \(x = r\cos \theta = (1 + \cos \theta)\cos \theta\) \(y = r\sin \theta = (1 + \cos \theta)\sin \theta\)

Step 2: Find the Distance

Find the distance \(d(\theta)\) from the origin to any point on the cardioid. We can use the Pythagorean theorem or the distance formula, which states that \(d(\theta) = \sqrt{x^2 + y^2}\). Substitute the expressions for \(x\) and \(y\) from step 1: \(d(\theta) = \sqrt{((1 + \cos \theta)\cos \theta)^2 + ((1 + \cos \theta)\sin \theta)^2}\)

Step 3: Simplify the Distance Expression

Simplify \(d(\theta)\) by applying trigonometric identities and properties: \(d(\theta) = \sqrt{(1 + \cos \theta)^2(\cos^2 \theta + \sin^2 \theta)}\) Since \(\cos^2 \theta + \sin^2 \theta = 1\), \(d(\theta) = (1 + \cos \theta)\)

Step 4: Set up the Integral for the Average Distance

Prepare the integral to represent the sum of the distances, over the domain of \(\theta\). As it is a cardioid, \(\theta\) ranges from 0 to 2\(\pi\). The average distance \(D_{avg}\) is the integral divided by the size of the interval (2\(\pi\)): \(D_{avg} = \frac{1}{2\pi}\int_{0}^{2\pi} d(\theta) d\theta = \frac{1}{2\pi}\int_{0}^{2\pi} (1 + \cos \theta) d\theta\)

Step 5: Calculate the Integral

Evaluate the integral from step 4 to obtain the average distance: \(D_{avg} = \frac{1}{2\pi} \left[\int_{0}^{2\pi}(1)d\theta + \int_{0}^{2\pi}(\cos \theta)d\theta\right]\) \(D_{avg} = \frac{1}{2\pi} \left[\theta \Big|_0^{2\pi} + \sin\theta \Big|_0^{2\pi} \right]\) \(D_{avg} = \frac{1}{2\pi} \left[(2\pi - 0) + (\sin(2\pi) - \sin(0)) \right]\)

Step 6: Simplify and Find the Average Distance

Simplify and find the average distance, using the result from step 5: \(D_{avg} = \frac{1}{2\pi} \left[2\pi\right]\) \(D_{avg} = 1\) The average distance between points within the cardioid \(r=1+\cos \theta\) and the origin is 1.