/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 46 Find the following average value... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 46

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

Short Answer

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

Step by step solution

01

Convert to Cartesian Coordinates

Convert the cardioid equation \(r=1+\cos \theta\) to Cartesian coordinates \((x,y)\) using polar-to-cartesian transformations: \(x = r\cos \theta = (1 + \cos \theta)\cos \theta\) \(y = r\sin \theta = (1 + \cos \theta)\sin \theta\)
02

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: \(d(\theta) = \sqrt{((1 + \cos \theta)\cos \theta)^2 + ((1 + \cos \theta)\sin \theta)^2}\)
03

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 \(\cos^2 \theta + \sin^2 \theta = 1\), \(d(\theta) = (1 + \cos \theta)\)
04

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\(\pi\)): \(D_{avg} = \frac{1}{2\pi}\int_{0}^{2\pi} d(\theta) d\theta = \frac{1}{2\pi}\int_{0}^{2\pi} (1 + \cos \theta) d\theta\)
05

Calculate the Integral

Evaluate the integral from step 4 to obtain the average distance: \(D_{avg} = \frac{1}{2\pi} \left[\int_{0}^{2\pi}(1)d\theta + \int_{0}^{2\pi}(\cos \theta)d\theta\right]\) \(D_{avg} = \frac{1}{2\pi} \left[\theta \Big|_0^{2\pi} + \sin\theta \Big|_0^{2\pi} \right]\) \(D_{avg} = \frac{1}{2\pi} \left[(2\pi - 0) + (\sin(2\pi) - \sin(0)) \right]\)
06

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.

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Most popular questions from this chapter

Use spherical coordinates to find the volume of the following solids. The solid cardioid of revolution \(D=\\{(\rho, \varphi, \theta): 0 \leq \rho \leq 1+\cos \varphi, 0 \leq \varphi \leq \pi, 0 \leq \theta \leq 2 \pi\\}\)

Evaluate the following integrals using the method of your choice. A sketch is helpful. $$\iint_{R} \sqrt{x^{2}+y^{2}} d A ; R=\\{(x, y): 0 \leq y \leq x \leq 1\\}$$

Mass from density Find the mass of the following solids with the given density functions. Note that density is described by the function \(f\) to avoid confusion with the radial spherical coordinate \(\rho\). The ball of radius 4 centered at the origin with a density \(f(\rho, \varphi, \theta)=1+\rho\)

For what values of \(p\) does the integral \(\iint_{R} \frac{d A}{\left(x^{2}+y^{2}\right)^{p}}\) exist in the following cases? a. \(R=\\{(r, \theta): 1 \leq r < \infty, 0 \leq \theta \leq 2 \pi\\}\) b. \(R=\\{(r, \theta): 0 \leq r \leq 1,0 \leq \theta \leq 2 \pi\\}\)

Changing order of integration If possible, write iterated integrals in cylindrical coordinates for the following regions in the specified orders. Sketch the region of integration. The solid above the cone \(z=r\) and below the sphere \(\rho=2,\) for \(z \geq 0,\) in the orders \(d z d r d \theta, d r d z d \theta,\) and \(d \theta d z d r\)
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