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 \( p, q \) if
\[
\int_{3}^{13} f(x) \, dx - \int_{3}^{6} f(x) \, dx = \int_{p}^{q} f(x) \, dx.
\]
(Give your answers as whole or exact numbers.)
---
**Student Response:**
\( p = 13 \)
*Incorrect*
\( q = 6 \)
*Incorrect*
---
**Explanation:**
The problem involves finding values \( p \) and \( q \) such that the equality holds for the given definite integrals.
Here is the breakdown of the problem:
1. The left side of the equation consists of two integrals: \( \int_{3}^{13} f(x) \, dx \) is the integral from 3 to 13, and \( \int_{3}^{6} f(x) \, dx \) is the integral from 3 to 6.
2. The result of these two integrals being subtracted is equivalent to the integral from 6 to 13:
\[
\int_{3}^{13} f(x) \, dx - \int_{3}^{6} f(x) \, dx = \int_{6}^{13} f(x) \, dx
\]
3. Therefore, for the equality to hold, the values of \( p \) and \( q \) should satisfy the integral:
\[
\int_{p}^{q} f(x) \, dx = \int_{6}^{13} f(x) \, dx
\]
4. Thus, \( p = 6 \) and \( q = 13 \).
This solution explains why the original answers were labeled incorrect and provides the correct values.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbe70156d-6630-4fa1-bd10-160edc6ae6ae%2F3142fb47-1a20-4ab9-bdc8-ccd09bb0a027%2Fn4377rm_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find \( p, q \) if
\[
\int_{3}^{13} f(x) \, dx - \int_{3}^{6} f(x) \, dx = \int_{p}^{q} f(x) \, dx.
\]
(Give your answers as whole or exact numbers.)
---
**Student Response:**
\( p = 13 \)
*Incorrect*
\( q = 6 \)
*Incorrect*
---
**Explanation:**
The problem involves finding values \( p \) and \( q \) such that the equality holds for the given definite integrals.
Here is the breakdown of the problem:
1. The left side of the equation consists of two integrals: \( \int_{3}^{13} f(x) \, dx \) is the integral from 3 to 13, and \( \int_{3}^{6} f(x) \, dx \) is the integral from 3 to 6.
2. The result of these two integrals being subtracted is equivalent to the integral from 6 to 13:
\[
\int_{3}^{13} f(x) \, dx - \int_{3}^{6} f(x) \, dx = \int_{6}^{13} f(x) \, dx
\]
3. Therefore, for the equality to hold, the values of \( p \) and \( q \) should satisfy the integral:
\[
\int_{p}^{q} f(x) \, dx = \int_{6}^{13} f(x) \, dx
\]
4. Thus, \( p = 6 \) and \( q = 13 \).
This solution explains why the original answers were labeled incorrect and provides the correct values.
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