3. f(x)=x+ln2, 3<<8
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Use Definition 2 to find an expression for the area under the graph of as a limit. Do not evaluate the limit.
Definition 2:
The area of the region that lies under the graph of the continuous function is the limit of the sum of the areas of approximating rectangles:
![2
Definition
The area A of the region S that lies under the graph of the continuous function f is
the limit of the sum of the areas of approximating rectangles:
A = lim Rn =
1-00
3
lim [f(x1) Ax + f(x₂) Ax+ ... + ƒ(Tn) Ax]
n→∞0
It can be proved that the limit in Definition 2 always exists, since we are assuming that f is
continuous. It can also be shown that we get the same value if we use left endpoints:
A = lim L₂ = lim [ƒ (x₁) Ax+ƒ (x₁) Ax+…+ ƒ (n-1) Ax]
17-00
004-26
In fact, instead of using left endpoints or right endpoints, we could take the height of the ith
rectangle to be the value of f at any number in the ith subinterval [i-1, ₁] We call the
numbers ₁,2,..., the sample points. Figure 13 shows approximating rectangles
when the sample points are not chosen to be endpoints. So a more general expression for
the area of S is](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc7a3cc6a-99d9-4a18-988a-5556d14d1218%2F1e737a79-3c52-45df-bffc-4c7bfb37cb9e%2Fftozdm_processed.png&w=3840&q=75)
Transcribed Image Text:2
Definition
The area A of the region S that lies under the graph of the continuous function f is
the limit of the sum of the areas of approximating rectangles:
A = lim Rn =
1-00
3
lim [f(x1) Ax + f(x₂) Ax+ ... + ƒ(Tn) Ax]
n→∞0
It can be proved that the limit in Definition 2 always exists, since we are assuming that f is
continuous. It can also be shown that we get the same value if we use left endpoints:
A = lim L₂ = lim [ƒ (x₁) Ax+ƒ (x₁) Ax+…+ ƒ (n-1) Ax]
17-00
004-26
In fact, instead of using left endpoints or right endpoints, we could take the height of the ith
rectangle to be the value of f at any number in the ith subinterval [i-1, ₁] We call the
numbers ₁,2,..., the sample points. Figure 13 shows approximating rectangles
when the sample points are not chosen to be endpoints. So a more general expression for
the area of S is
Transcribed Image Text:18. f(x) = x + lnx, 3≤x≤8
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