2) Consider the function g(x) = 10 - x² over the interval [1,3]. I want you to approximate the average value of this function over the interval [1, 3] by using four rectangles all of equal width. Use the midpoint of each subinterval to calculate the height of each rectangle. Give exact decimal answers. a) The area of these four rectangles is b) Find the approximated average value of this function over the given interval by using the area of the four rectangles.
2) Consider the function g(x) = 10 - x² over the interval [1,3]. I want you to approximate the average value of this function over the interval [1, 3] by using four rectangles all of equal width. Use the midpoint of each subinterval to calculate the height of each rectangle. Give exact decimal answers. a) The area of these four rectangles is b) Find the approximated average value of this function over the given interval by using the area of the four rectangles.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Exercise on Average Value of a Function**
**Problem Statement:**
Consider the function \( g(x) = 10 - x^2 \) over the interval \([1, 3]\).
We want you to approximate the *average value* of this function over the interval \([1, 3]\) by using four rectangles all of equal width. Use the **midpoint** of each subinterval to calculate the height of each rectangle. Provide exact decimal answers.
a) **The area of these four rectangles is** [__________].
b) **Find the approximated average value** of this function over the given interval by using the area of the four rectangles. [__________].
**Instructions:**
1. Divide the interval \([1, 3]\) into four equal subintervals.
2. Calculate the width of each rectangle.
3. Use the midpoint of each subinterval to find the height of the rectangles by evaluating the function \( g(x) \) at these midpoints.
4. Calculate the area of each rectangle and sum them to approximate the total area under the curve.
5. Divide the total area by the length of the interval to find the average value of the function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F02c8dc93-8b6f-4993-a985-0ce320a356a6%2F116cee23-9d45-4b50-ba65-8a23368ea242%2F55r4lx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise on Average Value of a Function**
**Problem Statement:**
Consider the function \( g(x) = 10 - x^2 \) over the interval \([1, 3]\).
We want you to approximate the *average value* of this function over the interval \([1, 3]\) by using four rectangles all of equal width. Use the **midpoint** of each subinterval to calculate the height of each rectangle. Provide exact decimal answers.
a) **The area of these four rectangles is** [__________].
b) **Find the approximated average value** of this function over the given interval by using the area of the four rectangles. [__________].
**Instructions:**
1. Divide the interval \([1, 3]\) into four equal subintervals.
2. Calculate the width of each rectangle.
3. Use the midpoint of each subinterval to find the height of the rectangles by evaluating the function \( g(x) \) at these midpoints.
4. Calculate the area of each rectangle and sum them to approximate the total area under the curve.
5. Divide the total area by the length of the interval to find the average value of the function.
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