Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Question
### Problem Description **Objective:** Set up an integral to find the area of the shaded region. Then, evaluate the integral to determine the area. ### Explanation of the Graph **Graph Overview:** - The graph displays the horizontal and vertical axes labeled as \(x\) and \(y\), respectively. - The plot includes three main curves and lines: 1. **Blue Curve:** \(x = y^2 - 2\) 2. **Red Curve:** \(x = e^y\) 3. **Horizontal Lines:** \(y = 1\) and \(y = -1\) - The shaded region lies between these curves and lines, constrained vertically between \(y = 1\) and \(y = -1\). **Boundaries of the Shaded Region:** - **Left Boundary:** Defined by \(x = y^2 - 2\) - **Right Boundary:** Defined by \(x = e^y\) - **Top Boundary:** \(y = 1\) - **Bottom Boundary:** \(y = -1\) ### Setting Up the Integral To find the area of the shaded region, express it as an integral with respect to \(y\). The limits of integration will be from \(y = -1\) to \(y = 1\). **Integral Expression:** \[ \int_{-1}^{1} \left(e^y - (y^2 - 2)\right) \, dy \] ### Evaluating the Integral To compute the area, evaluate the integral: \[ \int_{-1}^{1} \left(e^y - y^2 + 2\right) \, dy \] This integral represents the difference in the area between the exponential function \(e^y\) and the parabola \(y^2 - 2\), over the specified range of \(y\). ### Summary This problem covers topics such as setting up integrals, understanding the graph of a function, and calculating area under a curve. By computing the integral, you will find the precise area of the shaded region between the given curves and lines.
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Transcribed Image Text:### Problem Description **Objective:** Set up an integral to find the area of the shaded region. Then, evaluate the integral to determine the area. ### Explanation of the Graph **Graph Overview:** - The graph displays the horizontal and vertical axes labeled as \(x\) and \(y\), respectively. - The plot includes three main curves and lines: 1. **Blue Curve:** \(x = y^2 - 2\) 2. **Red Curve:** \(x = e^y\) 3. **Horizontal Lines:** \(y = 1\) and \(y = -1\) - The shaded region lies between these curves and lines, constrained vertically between \(y = 1\) and \(y = -1\). **Boundaries of the Shaded Region:** - **Left Boundary:** Defined by \(x = y^2 - 2\) - **Right Boundary:** Defined by \(x = e^y\) - **Top Boundary:** \(y = 1\) - **Bottom Boundary:** \(y = -1\) ### Setting Up the Integral To find the area of the shaded region, express it as an integral with respect to \(y\). The limits of integration will be from \(y = -1\) to \(y = 1\). **Integral Expression:** \[ \int_{-1}^{1} \left(e^y - (y^2 - 2)\right) \, dy \] ### Evaluating the Integral To compute the area, evaluate the integral: \[ \int_{-1}^{1} \left(e^y - y^2 + 2\right) \, dy \] This integral represents the difference in the area between the exponential function \(e^y\) and the parabola \(y^2 - 2\), over the specified range of \(y\). ### Summary This problem covers topics such as setting up integrals, understanding the graph of a function, and calculating area under a curve. By computing the integral, you will find the precise area of the shaded region between the given curves and lines.
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