Fundemental Theorem of Integral Calculus Suppose f: [a, b] → R is differen- tiable on [a, b] and ƒ' € R(x) on [a,b]. Then [" f'dx = f(b) – ƒ(a). = Define ƒ : [0,2] → R by f(x) = 2x - r² for 0 < x < 1 and f(x) (x-2)² for 1 < x < 2. Prove that f is integrable on [0,2] and find the integral of f over [0,2]. Don not use Theorem 5.10, but rather find the integral by methods similar to those used in the proof of Theorem 5.8.
Fundemental Theorem of Integral Calculus Suppose f: [a, b] → R is differen- tiable on [a, b] and ƒ' € R(x) on [a,b]. Then [" f'dx = f(b) – ƒ(a). = Define ƒ : [0,2] → R by f(x) = 2x - r² for 0 < x < 1 and f(x) (x-2)² for 1 < x < 2. Prove that f is integrable on [0,2] and find the integral of f over [0,2]. Don not use Theorem 5.10, but rather find the integral by methods similar to those used in the proof of Theorem 5.8.
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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![Fundemental Theorem of Integral Calculus Suppose f: [a, b] → R is differen-
tiable on [a, b] and ƒ' € R(x) on [a,b]. Then
[" f'dx = f(b) – ƒ(a).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2eed1293-78aa-42e8-9d83-b686fedc1149%2Fc4244eb2-9780-4b7d-94aa-42dfd5129285%2Foao7wt_processed.png&w=3840&q=75)
Transcribed Image Text:Fundemental Theorem of Integral Calculus Suppose f: [a, b] → R is differen-
tiable on [a, b] and ƒ' € R(x) on [a,b]. Then
[" f'dx = f(b) – ƒ(a).
![=
Define ƒ : [0,2] → R by f(x) = 2x - r² for 0 < x < 1 and f(x) (x-2)² for
1 < x < 2. Prove that f is integrable on [0,2] and find the integral of f over [0,2].
Don not use Theorem 5.10, but rather find the integral by methods similar to those
used in the proof of Theorem 5.8.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2eed1293-78aa-42e8-9d83-b686fedc1149%2Fc4244eb2-9780-4b7d-94aa-42dfd5129285%2Fmjkw76_processed.png&w=3840&q=75)
Transcribed Image Text:=
Define ƒ : [0,2] → R by f(x) = 2x - r² for 0 < x < 1 and f(x) (x-2)² for
1 < x < 2. Prove that f is integrable on [0,2] and find the integral of f over [0,2].
Don not use Theorem 5.10, but rather find the integral by methods similar to those
used in the proof of Theorem 5.8.
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