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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![**Problem Statement:**
Find the area of the region enclosed between the curves \(f(x) = 0.7x^2 + 4\) and \(g(x) = x\), within the interval \(x = -6\) to \(x = 5\).
**Graph Description:**
- The graph displays two functions: \(f(x) = 0.7x^2 + 4\) (in blue) and \(g(x) = x\) (in red).
- The blue curve (\(f(x)\)) represents a parabola opening upwards with its vertex above the x-axis.
- The red line (\(g(x)\)) represents a straight line with a slope of 1.
- Vertical lines are plotted at \(x = -6\) and \(x = 5\) to mark the boundaries of the interval we're interested in.
**Graph Analysis:**
- The parabola \(f(x)\) intersects the y-axis above the line \(g(x)\), indicating they enclose an area between the two curves over the interval.
- The region of interest is the area between these two curves from \(x = -6\) to \(x = 5\).
**Area Calculation:**
To find the area \(A\) between the functions, you can use the integral of the difference between the functions over the given interval:
\[ A = \int_{-6}^{5} \left[ f(x) - g(x) \right] dx \]
Substitute the given functions into the integral:
\[ A = \int_{-6}^{5} \left[ (0.7x^2 + 4) - x \right] dx \]
\[ A = \int_{-6}^{5} \left[ 0.7x^2 + 4 - x \right] dx \]
Evaluate the integral to find the area.
**Input Box:**
An input box labeled "Area =" is provided for students or users to enter the calculated area after they have evaluated the integral.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F686df84f-2572-4ea8-99e4-a6aa8d03e207%2Fa22455fe-cbfe-4310-a52b-e37b4af2bb82%2Fi9fbvuh_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the area of the region enclosed between the curves \(f(x) = 0.7x^2 + 4\) and \(g(x) = x\), within the interval \(x = -6\) to \(x = 5\).
**Graph Description:**
- The graph displays two functions: \(f(x) = 0.7x^2 + 4\) (in blue) and \(g(x) = x\) (in red).
- The blue curve (\(f(x)\)) represents a parabola opening upwards with its vertex above the x-axis.
- The red line (\(g(x)\)) represents a straight line with a slope of 1.
- Vertical lines are plotted at \(x = -6\) and \(x = 5\) to mark the boundaries of the interval we're interested in.
**Graph Analysis:**
- The parabola \(f(x)\) intersects the y-axis above the line \(g(x)\), indicating they enclose an area between the two curves over the interval.
- The region of interest is the area between these two curves from \(x = -6\) to \(x = 5\).
**Area Calculation:**
To find the area \(A\) between the functions, you can use the integral of the difference between the functions over the given interval:
\[ A = \int_{-6}^{5} \left[ f(x) - g(x) \right] dx \]
Substitute the given functions into the integral:
\[ A = \int_{-6}^{5} \left[ (0.7x^2 + 4) - x \right] dx \]
\[ A = \int_{-6}^{5} \left[ 0.7x^2 + 4 - x \right] dx \]
Evaluate the integral to find the area.
**Input Box:**
An input box labeled "Area =" is provided for students or users to enter the calculated area after they have evaluated the integral.
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