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...
Related questions
Question
![The image contains a mathematical statement involving definite integrals. The statement reads:
"If
\[ \int_{2}^{8} f(x) \, dx = -43, \]
then
\[ \int_{8}^{2} f(t) \, dt = \]"
To provide some context for educational purposes, we can add an explanation:
### Explanation:
A definite integral of a function \( f(x) \) over the interval \([a, b]\) is represented as \( \int_{a}^{b} f(x) \, dx \). This integral calculates the net area under the curve of \( f(x) \) between \( x = a \) and \( x = b \).
In this problem, we are given that the definite integral of \( f(x) \) from \( x = 2 \) to \( x = 8 \) is \(-43\):
\[ \int_{2}^{8} f(x) \, dx = -43. \]
We need to find the value of the definite integral of \( f(t) \) from \( t = 8 \) to \( t = 2 \):
\[ \int_{8}^{2} f(t) \, dt. \]
To compute this, we use the property of definite integrals that states reversing the limits of integration changes the sign of the integral:
\[ \int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx. \]
Applying this property:
\[ \int_{8}^{2} f(t) \, dt = -\int_{2}^{8} f(t) \, dt. \]
Given that \( \int_{2}^{8} f(x) \, dx = -43 \),
\[ \int_{8}^{2} f(t) \, dt = -(-43) = 43. \]
Therefore, the answer is:
\[ \int_{8}^{2} f(t) \, dt = 43. \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F139cb5f2-923a-4627-93ad-b9a40e985e19%2F437676f1-77b4-4c96-94ca-8f68629a5076%2Fzql8s8s.png&w=3840&q=75)
Transcribed Image Text:The image contains a mathematical statement involving definite integrals. The statement reads:
"If
\[ \int_{2}^{8} f(x) \, dx = -43, \]
then
\[ \int_{8}^{2} f(t) \, dt = \]"
To provide some context for educational purposes, we can add an explanation:
### Explanation:
A definite integral of a function \( f(x) \) over the interval \([a, b]\) is represented as \( \int_{a}^{b} f(x) \, dx \). This integral calculates the net area under the curve of \( f(x) \) between \( x = a \) and \( x = b \).
In this problem, we are given that the definite integral of \( f(x) \) from \( x = 2 \) to \( x = 8 \) is \(-43\):
\[ \int_{2}^{8} f(x) \, dx = -43. \]
We need to find the value of the definite integral of \( f(t) \) from \( t = 8 \) to \( t = 2 \):
\[ \int_{8}^{2} f(t) \, dt. \]
To compute this, we use the property of definite integrals that states reversing the limits of integration changes the sign of the integral:
\[ \int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx. \]
Applying this property:
\[ \int_{8}^{2} f(t) \, dt = -\int_{2}^{8} f(t) \, dt. \]
Given that \( \int_{2}^{8} f(x) \, dx = -43 \),
\[ \int_{8}^{2} f(t) \, dt = -(-43) = 43. \]
Therefore, the answer is:
\[ \int_{8}^{2} f(t) \, dt = 43. \]
Expert Solution
Step 1
Given
By the property of definite integral, we have
Step 2
Therefore, we have
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 4 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning