/*! 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 76 Surface Area Find the area of th... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 76

Surface Area Find the area of the surface formed by revolving the graph of \(y=2 e^{-x}\) on the interval \([0, \infty)\) about the \(x\) -axis.

Short Answer

Expert verified
The surface area of the surface formed by revolving the graph around the \(x\)-axis is \(2\pi\) units square.

Step by step solution

01

Identify the Form of Surface Area for Revolution

To find the surface area, we'll use the formula for the surface area \(S\) formed by revolving a curve \(y=f(x)\), defined on the interval \([a, b]\), around the \(x\)-axis:\[ S = 2\pi \int_a^b y \sqrt{1+ (f'(x))^2} dx \]In this problem, the given function is \(y=2e^{-x}\) and \(y'\) (the derivative of the function with respect to \(x\)) is equal to \(-2e^{-x}\). Thus, we replace \(f(x)\) and \(f'(x)\) in the formula above with this information.
02

Substitution into the Formula

Substitute \(y\), \(f'(x)\), \(a\) and \(b\) into the surface area formula. Here \(a=0\) and \(b=鈭瀄), so we get:\[ S = 2\pi \int_0^鈭� 2e^{-x}\sqrt{1+ (-2e^{-x})^2} dx \]
03

Simplification of the Integral

After simplification, we obtain\[ S = 2\pi \int_0^鈭� e^{-x} dx \]
04

Evaluate the Improper Integral

The integral of \(e^{-x}\) is \(-e^{-x}\), so by plugging in values we get \[ S = 2\pi (-e^{-鈭瀩 + e^{-0}) \]
05

Simplify the Final Answer

As \(e^{-鈭瀩 = 0\) we find that\[ S = 2\pi(0 + 1) = 2\pi \]

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