Find the area under the graph of f over the interval [1,9]. 8x + 7, f(x) = 5 111-2. for x = 8 for x> 8 The area is. (Type an integer or a simplified fraction.) www
Find the area under the graph of f over the interval [1,9]. 8x + 7, f(x) = 5 111-2. for x = 8 for x> 8 The area is. (Type an integer or a simplified fraction.) www
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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![**Math Problem: Calculating Area Under a Piecewise Function**
**Problem Statement:**
Find the area under the graph of \( f \) over the interval \([1,9]\).
**Function Definition:**
\[
f(x) =
\begin{cases}
8x + 7, & \text{for } x \leq 8 \\
111 - \frac{5}{2}x, & \text{for } x > 8
\end{cases}
\]
**Solution:**
To find the area under the curve defined by \( f(x) \) over the interval \([1,9]\), we need to integrate each piece of the piecewise function over its respective subinterval and sum the results.
1. **Calculate for \( x \leq 8 \):**
Evaluate the integral of \( 8x + 7 \) from 1 to 8.
2. **Calculate for \( x > 8 \):**
Evaluate the integral of \( 111 - \frac{5}{2}x \) from 8 to 9.
3. **Sum the Areas:**
Add the results of the two integrals to get the total area.
**Input the Area:**
The area is \(\boxed{ }\) (Type an integer or a simplified fraction.)
**Additional Resources:**
- [View an example]
- [Get more help]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1ecf61ef-f650-4520-87fd-97345184f757%2Ffaef2011-8d0c-4b7e-9837-6c3436c4dbea%2Fqnvaiw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Math Problem: Calculating Area Under a Piecewise Function**
**Problem Statement:**
Find the area under the graph of \( f \) over the interval \([1,9]\).
**Function Definition:**
\[
f(x) =
\begin{cases}
8x + 7, & \text{for } x \leq 8 \\
111 - \frac{5}{2}x, & \text{for } x > 8
\end{cases}
\]
**Solution:**
To find the area under the curve defined by \( f(x) \) over the interval \([1,9]\), we need to integrate each piece of the piecewise function over its respective subinterval and sum the results.
1. **Calculate for \( x \leq 8 \):**
Evaluate the integral of \( 8x + 7 \) from 1 to 8.
2. **Calculate for \( x > 8 \):**
Evaluate the integral of \( 111 - \frac{5}{2}x \) from 8 to 9.
3. **Sum the Areas:**
Add the results of the two integrals to get the total area.
**Input the Area:**
The area is \(\boxed{ }\) (Type an integer or a simplified fraction.)
**Additional Resources:**
- [View an example]
- [Get more help]
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