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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Question
![**Evaluating a Definite Integral**
In this exercise, we aim to evaluate the following definite integral:
\[
\int_{0}^{1} (8 + \cosh \, t) \, dt
\]
Here is the breakdown of the components:
- The integral is evaluated from the lower limit \( t = 0 \) to the upper limit \( t = 1 \).
- The integrand is \( 8 + \cosh \, t \), where \( \cosh \) represents the hyperbolic cosine function.
### Steps to Evaluate the Integral:
1. **Integrate each term separately:**
- For the constant term \( 8 \):
\[
\int 8 \, dt = 8t + C
\]
- For the hyperbolic cosine term \( \cosh \, t \):
\[
\int \cosh t \, dt = \sinh t + C
\]
2. **Combine the results of the separate integrations:**
\[
\int (8 + \cosh t) \, dt = 8t + \sinh t + C
\]
3. **Evaluate the antiderivative at the bounds of the integral, from \( t = 0 \) to \( t = 1 \):**
\[
\left. (8t + \sinh t) \right|_0^1 = \left(8(1) + \sinh(1)\right) - \left(8(0) + \sinh(0)\right)
\]
4. **Simplify the expression:**
\[
8 + \sinh(1) - 0 = 8 + \sinh(1)
\]
Hence, the evaluated integral is:
\[
8 + \sinh(1)
\]
This result can be further approximated using a calculator to find the numerical value if needed. For accurate approximation, you can use the value of \( \sinh(1) \approx 1.175 \).
Therefore, \( 8 + \sinh(1) \approx 9.175 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5a900a57-4543-4439-89eb-9eb30a858e98%2Fd0dbd1f1-b6d6-4553-8955-317d2a8016c8%2Fywdrb1c.jpeg&w=3840&q=75)
Transcribed Image Text:**Evaluating a Definite Integral**
In this exercise, we aim to evaluate the following definite integral:
\[
\int_{0}^{1} (8 + \cosh \, t) \, dt
\]
Here is the breakdown of the components:
- The integral is evaluated from the lower limit \( t = 0 \) to the upper limit \( t = 1 \).
- The integrand is \( 8 + \cosh \, t \), where \( \cosh \) represents the hyperbolic cosine function.
### Steps to Evaluate the Integral:
1. **Integrate each term separately:**
- For the constant term \( 8 \):
\[
\int 8 \, dt = 8t + C
\]
- For the hyperbolic cosine term \( \cosh \, t \):
\[
\int \cosh t \, dt = \sinh t + C
\]
2. **Combine the results of the separate integrations:**
\[
\int (8 + \cosh t) \, dt = 8t + \sinh t + C
\]
3. **Evaluate the antiderivative at the bounds of the integral, from \( t = 0 \) to \( t = 1 \):**
\[
\left. (8t + \sinh t) \right|_0^1 = \left(8(1) + \sinh(1)\right) - \left(8(0) + \sinh(0)\right)
\]
4. **Simplify the expression:**
\[
8 + \sinh(1) - 0 = 8 + \sinh(1)
\]
Hence, the evaluated integral is:
\[
8 + \sinh(1)
\]
This result can be further approximated using a calculator to find the numerical value if needed. For accurate approximation, you can use the value of \( \sinh(1) \approx 1.175 \).
Therefore, \( 8 + \sinh(1) \approx 9.175 \).
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