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Find the area of the shaded region below. y A y = Vx+2 x = 2 1 y%3D x+1

Find the area of the shaded region below. y A y = Vx+2 x = 2 1 y%3D x+1

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Title: Calculating the Area of a Shaded Region Between Curves** --- **Objective:** Find the area of the shaded region bounded by the given curves. **Graph Description:** The graph consists of a Cartesian coordinate system with the following elements: 1. **Red Curve (Upper Boundary):** - Equation: \( y = \sqrt{x + 2} \) - This curve represents a portion of a parabola shifted upward with a square root function, indicating it opens to the right. 2. **Blue Curve (Lower Boundary):** - Equation: \( y = \frac{1}{x+1} \) - This is a rational function representing a hyperbola, with a vertical asymptote at \( x = -1 \) and a horizontal asymptote at \( y = 0 \). 3. **Vertical Line (Right Boundary):** - Equation: \( x = 2 \) - This line acts as a vertical bound on the right. **Shaded Region:** - The shaded area represents the region between the curves \( y = \sqrt{x + 2} \) and \( y = \frac{1}{x+1} \), bounded to the left by the y-axis and to the right by the vertical line \( x = 2 \). - It is essential to determine where these curves intersect, calculate the definite integral of their difference from the left to the right boundary (where \( x = 2 \)), and find the total area. **Steps to Calculate the Area:** 1. **Find Intersection Points:** - Solve the equations \( \sqrt{x + 2} = \frac{1}{x+1} \) to find the x-coordinates of intersection points. 2. **Calculate the Definite Integral:** - Integrate the difference of the two functions (\( \sqrt{x + 2} - \frac{1}{x+1} \)) with respect to \( x \) over the interval defined by the intersection points and the line \( x = 2 \). This is a classic problem in calculus involving integration to find the area between curves.
Transcribed Image Text:**Title: Calculating the Area of a Shaded Region Between Curves** --- **Objective:** Find the area of the shaded region bounded by the given curves. **Graph Description:** The graph consists of a Cartesian coordinate system with the following elements: 1. **Red Curve (Upper Boundary):** - Equation: \( y = \sqrt{x + 2} \) - This curve represents a portion of a parabola shifted upward with a square root function, indicating it opens to the right. 2. **Blue Curve (Lower Boundary):** - Equation: \( y = \frac{1}{x+1} \) - This is a rational function representing a hyperbola, with a vertical asymptote at \( x = -1 \) and a horizontal asymptote at \( y = 0 \). 3. **Vertical Line (Right Boundary):** - Equation: \( x = 2 \) - This line acts as a vertical bound on the right. **Shaded Region:** - The shaded area represents the region between the curves \( y = \sqrt{x + 2} \) and \( y = \frac{1}{x+1} \), bounded to the left by the y-axis and to the right by the vertical line \( x = 2 \). - It is essential to determine where these curves intersect, calculate the definite integral of their difference from the left to the right boundary (where \( x = 2 \)), and find the total area. **Steps to Calculate the Area:** 1. **Find Intersection Points:** - Solve the equations \( \sqrt{x + 2} = \frac{1}{x+1} \) to find the x-coordinates of intersection points. 2. **Calculate the Definite Integral:** - Integrate the difference of the two functions (\( \sqrt{x + 2} - \frac{1}{x+1} \)) with respect to \( x \) over the interval defined by the intersection points and the line \( x = 2 \). This is a classic problem in calculus involving integration to find the area between curves.
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