Set up the definite integral required to find the area of the region between the graph of y = 12 – z? and y = 21x + 102 over the interval –1 < x < 2. da
Set up the definite integral required to find the area of the region between the graph of y = 12 – z? and y = 21x + 102 over the interval –1 < x < 2. da
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 Between Two Curves Using Definite Integrals**
**Problem Statement:**
Set up the definite integral required to find the area of the region between the graph of \( y = 12 - x^2 \) and \( y = 21x + 102 \) over the interval \( -1 \leq x \leq 2 \).
**Visual Representation:**
There is a rectangular box, intended for input of the integral expression, alongside an integral sign \( \int \) and the differential \( dx \).
**Explanation:**
To find the area between these two curves, you need to set up and evaluate the definite integral of the difference between the two functions over the given interval. The area \( A \) can be expressed as:
\[
A = \int_{-1}^{2} \left( (21x + 102) - (12 - x^2) \right) \, dx
\]
The expression within the integral represents the top function minus the bottom function over the interval from \( x = -1 \) to \( x = 2 \).
**Steps:**
1. Identify the functions: \( y_1 = 12 - x^2 \) and \( y_2 = 21x + 102 \).
2. Determine the interval: \( -1 \leq x \leq 2 \).
3. Set up the definite integral of the difference \( y_2 - y_1 \) over the interval.
This setup involves determining which graph is on top and which is on the bottom within the interval, and integrating the difference of these functions with respect to \( x \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6cf09b2a-e92e-430b-8152-8153c73e6dd1%2F3f56e7d2-99b4-422b-a30f-95a3c2a4a128%2Fwexx0i9_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Calculating the Area Between Two Curves Using Definite Integrals**
**Problem Statement:**
Set up the definite integral required to find the area of the region between the graph of \( y = 12 - x^2 \) and \( y = 21x + 102 \) over the interval \( -1 \leq x \leq 2 \).
**Visual Representation:**
There is a rectangular box, intended for input of the integral expression, alongside an integral sign \( \int \) and the differential \( dx \).
**Explanation:**
To find the area between these two curves, you need to set up and evaluate the definite integral of the difference between the two functions over the given interval. The area \( A \) can be expressed as:
\[
A = \int_{-1}^{2} \left( (21x + 102) - (12 - x^2) \right) \, dx
\]
The expression within the integral represents the top function minus the bottom function over the interval from \( x = -1 \) to \( x = 2 \).
**Steps:**
1. Identify the functions: \( y_1 = 12 - x^2 \) and \( y_2 = 21x + 102 \).
2. Determine the interval: \( -1 \leq x \leq 2 \).
3. Set up the definite integral of the difference \( y_2 - y_1 \) over the interval.
This setup involves determining which graph is on top and which is on the bottom within the interval, and integrating the difference of these functions with respect to \( x \).
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