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Chapter 8: Problem 52

Give the integral formulas for the area of the surface of revolution formed when the graph of \(r=f(\theta)\) is revolved about (a) the \(x\) -axis and (b) the \(y\) -axis.

Short Answer

Expert verified
For revolving around the x-axis, the integral is \(S = 2\pi\int_a^b f(\theta) | \sin\theta | \sqrt{1+ \left(\frac{f'(\theta) \cos\theta - f(\theta) \sin\theta}{f(\theta) \cos\theta}\right)^2} f(\theta) \cos\theta\ d\theta\). For revolving around y-axis, the integral is \(S = 2\pi\int_c^d f(\theta) | \cos\theta | \sqrt{1+ \left(\frac{f'(\theta) \sin\theta + f(\theta) \cos\theta}{f(\theta) \sin\theta}\right)^2} f(\theta) \sin\theta\ d\theta\).

Step by step solution

01

Identify Polar Coordinates

In polar coordinates, the function is \(r = f(\theta)\). A point is defined by the distance \(r\) from the origin and the angle \(\theta\) it makes with the x-axis.
02

Convert To Cartesian Coordinates For Revolving Around the X-Axis

The conversion from polar to Cartesian coordinates results in: \(x = r \cos\theta = f(\theta) \cos\theta\) and \(y = r \sin\theta = f(\theta) \sin\theta\). The surface area \(S\) we are looking for is described by the formula: \(S = 2\pi\int_a^b y\sqrt{1+\left(\frac{dy}{dx}\right)^2}\ dx\), but we need to find the derivative in terms of \(\theta\). Therefore we differentiate \(y\) with respect to \(x\), apply the resulting derivative to the surface area formula, and then achieve the integral.
03

Formulate the Integral for Revolving Around the X-Axis

After the differentiation, we find the integral that gives us the surface area for the revolution around the x-axis: \(S = 2\pi\int_a^b f(\theta) | \sin\theta | \sqrt{1+ \left(\frac{f'(\theta) \cos\theta - f(\theta) \sin\theta}{f(\theta) \cos\theta}\right)^2} f(\theta) \cos\theta\ d\theta\).
04

Convert To Cartesian Coordinates For Revolving Around the Y-Axis

Again, the conversion from polar to Cartesian coordinates results in: \(x = r \cos\theta = f(\theta) \cos\theta\) and \(y = r \sin\theta = f(\theta) \sin\theta\). Now we refer to the surface area formula with respect to the y-axis: \(S = 2\pi\int_c^d x\sqrt{1+\left(\frac{dx}{dy}\right)^2}\ dy\), where we need to find the derivative in terms of \(\theta\). So we differentiate \(x\) with respect to \(y\), apply the resulting derivative to the surface area formula, and then achieve the integral.
05

Formulate the Integral for Revolving Around the Y-Axis

With the derivative, we get the integral that gives us the surface area for revolution around the y-axis: \(S = 2\pi\int_c^d f(\theta) | \cos\theta | \sqrt{1+ \left(\frac{f'(\theta) \sin\theta + f(\theta) \cos\theta}{f(\theta) \sin\theta}\right)^2} f(\theta) \sin\theta\ d\theta\).

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Most popular questions from this chapter

Writing Consider the polar equation \(r=\frac{4}{1+e \sin \theta} .\) (a) Use a graphing utility to graph the equation for \(e=0.1\), \(e=0.25, e=0.5, e=0.75,\) and \(e=0.9 .\) Identify the conic and discuss the change in its shape as \(e \rightarrow 1^{-}\) and \(e \rightarrow 0^{+}\) (b) Use a graphing utility to graph the equation for \(e=1\). Identify the conic. (c) Use a graphing utility to graph the equation for \(e=1.1\), \(e=1.5,\) and \(e=2 .\) Identify the conic and discuss the change in its shape as \(e \rightarrow 1^{+}\) and \(e \rightarrow \infty\).

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In Exercises \(17-20,\) use a graphing utility to graph the polar equation. Identify the graph. \(r=\frac{3}{-4+2 \sin \theta}\)

Use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. $$ \text { Cycloid: } x=2(\theta-\sin \theta), \quad y=2(1-\cos \theta) $$
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