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Essential Calculus: Early Transcendentals
Found in: Page 369
Essential Calculus: Early Transcendentals

Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

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Short Answer

Find the area of the shaped region.

The shaded area is\(e - \frac{1}{e} + \frac{{10}}{3}\)square units.

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Given information

Consider two continuous functions \(f(y)\)and \(g(y)\)such that \(f(y) \ge g(y)\)for \(y \in (c,d).\)

The area bounded between \(x = f(y),x = g(y)\)and the horizontal lines \(y = c,d\)is given by

\(A = \int_c^d f (y) - g(y)dy\)

Finding the area of shaped region

On the interval \(y \in \left( { - 1,1} \right)\), we can observe that \({e^y} \ge {y^2} - 2\), on the graph

As a result, the shaded area is

\(\begin{aligned}{l}\int\limits_{ - 1}^1 {{e^y} - \left( {{y^2} - 2} \right)} dy\\ = \int\limits_{ - 1}^1 {{e^y} - } {y^2} + 2dy\\ = \left( {{e^y} - \frac{{{y^{2 + 1}}}}{{2 + 1}} + 2y} \right)_{ - 1}^1\end{aligned}\)

\( = \left( {{e^1} - \frac{{{1^3}}}{3} + 2\left( 1 \right)} \right) - \left( {{e^{ - 1}} - \frac{{{{( - 1)}^3}}}{3} + 2\left( { - 1} \right)} \right)\)

\( = \left( {e - \frac{1}{3} + 2} \right) - \left( {{e^{ - 1}} + \frac{1}{3} - 2} \right)\)

\( = e - \frac{1}{e} - \frac{2}{3} + 4\)

The shaded area is\(e - \frac{1}{e} + \frac{{10}}{3}\)square units.

Most popular questions for Math Textbooks

Question: Suppose that \(0 < c < \frac{\pi }{2}\). For what value of \(c\) is the area of the region enclosed by the curves \(y = \cos x,y = \cos (x - c)\), and \(x = 0\) equal to the area of the region enclosed by the curves \(y = \cos (x - c),x = \pi \), and \(y = 0\) ?

Graph the region between the curves and use your calculator to compute the area correct to five decimal places.

\(y = {\tan ^2}x,\quad y = \sqrt x \)

Find the area of the crescent-shaped region (called a lune) bounded by arcs of circles with radii and (see the figure)

(a) To set up: an integral function for the volume of the solid obtained by rotating the region bounded by the given curve.

(b)To evaluate: The integral function.

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places.

\(y = {e^{ - {x^2}}},y = 0,x = - 1,x = 1\)

a) About the\(x - \) axis

b) About \(y = - 1\)
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