/*! 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 58 A torus is formed by revolving t... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 58

A torus is formed by revolving the graph of \((x-1)^{2}+y^{2}=1\) about the \(y\) -axis. Find the surface area of the torus.

Short Answer

Expert verified
The surface area of the torus is achieved by performing the calculation in Step 3. After integrating, one will get the surface area of the torus. It requires some familiarity with integral calculus to solve but is straightforward once the necessary expressions have been derived.

Step by step solution

01

Identify the Function

The function to be revolved is the circle \((x-1)^{2}+y^{2}=1\). Express it in terms of \(x\) and \(y\) as \(y=\sqrt{1-(x-1)^{2}}\). We only consider the positive root because we're revolving around the y-axis and we are above the x-axis.
02

Calculate the Derivative

Compute the derivative of \(y\) with respect to \(x\), denoted as \(y'(x)\). Differentiating \(y=\sqrt{1-(x-1)^{2}}\) yields \(y'(x) = -\frac{(x-1)}{\sqrt{1-(x-1)^{2}}}\). The derivative will be used in the formula for the surface area of a solid of revolution.
03

Calculate the Surface Area

Substitute \(y\) and \(y'(x)\) into the formula for the surface area of a solid of revolution, and evaluate the integral from \(-1\) to \(1\). This gives us the surface area \(2\pi\int_{-1}^{1} \sqrt{1-(x-1)^{2}} \sqrt{1 +\left [ \frac{(x-1)}{\sqrt{1-(x-1)^{2}}} \right ]^{2}} dx\). The integral can be simplified to \(2\pi\int_{-1}^{1} \sqrt{1-(x-1)^{2}} \sqrt{2} dx\). Hence calculate the integral to obtain the surface area of the torus.

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Most popular questions from this chapter

A sphere of radius \(r\) is generated by revolving the graph of \(y=\sqrt{r^{2}-x^{2}}\) about the \(x\) -axis. Verify that the surface area of the sphere is \(4 \pi r^{2}\).

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