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
100%
Evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2.
181.
16
∫ (dt / t^4)
1
185.
(π/4)
∫ sec θ t and θ
0
![b The distribution of incon x
O Homework: Section 5.4
E 5.3 The Fundamental Th
b Answered: exercises, use x
O Selena Gomez - Sam
O Qun fine settimana - 17 x
+
A https://openstax.org/books/calculus-volume-1/pages/5-3-the-fundamental-theorem-of-calculus#fs-id1170572376509
< Calculus Volume 1
5.3 The Fundamental Theorem of Calculus
E Table of contents
Search this book
9 My highlights
B Print
The Fundamental Theorem of Calculus, Part 2
If f is continuous over the interval [a, b] and F (x) is any antiderivative of f (x), then
| f (æ) dæ = F (b) – F (a).
5.17
We often see the notation F (x)I to denote the expression F (b) – F (a). We use this vertical bar and associated limits a and b
to indicate that we should evaluate the function F (x) at the upper limit (in this case, b), and subtract the value of the function
F (x) evaluated at the lower limit (in this case, a).
The Fundamental Theorem of Calculus, Part 2 (also known as the evaluation theorem) states that if we can find an antiderivative
for the integrand, then we can evaluate the definite integral by evaluating the antiderivative at the endpoints of the interval and
subtracting.
Proof
Let P = {x;},i = 0, 1,..., n be a regular partition of [a, b] . Then, we can write
F (b) – F (a) = F (x„) – F (xo)
= [F (x„) – F (x„-1)] + [F (x,-1) – F (x„ 2)] + ...+ [F (x1) – F (xo)]
=2 F (z:) – F (z;-1) -
M
•A 3:04](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d6a2302-be00-405d-bbf6-4b5c194aa6b1%2F7a18b58e-008b-42e4-ab88-6a2723ac1213%2Fafngpl_processed.png&w=3840&q=75)
Transcribed Image Text:b The distribution of incon x
O Homework: Section 5.4
E 5.3 The Fundamental Th
b Answered: exercises, use x
O Selena Gomez - Sam
O Qun fine settimana - 17 x
+
A https://openstax.org/books/calculus-volume-1/pages/5-3-the-fundamental-theorem-of-calculus#fs-id1170572376509
< Calculus Volume 1
5.3 The Fundamental Theorem of Calculus
E Table of contents
Search this book
9 My highlights
B Print
The Fundamental Theorem of Calculus, Part 2
If f is continuous over the interval [a, b] and F (x) is any antiderivative of f (x), then
| f (æ) dæ = F (b) – F (a).
5.17
We often see the notation F (x)I to denote the expression F (b) – F (a). We use this vertical bar and associated limits a and b
to indicate that we should evaluate the function F (x) at the upper limit (in this case, b), and subtract the value of the function
F (x) evaluated at the lower limit (in this case, a).
The Fundamental Theorem of Calculus, Part 2 (also known as the evaluation theorem) states that if we can find an antiderivative
for the integrand, then we can evaluate the definite integral by evaluating the antiderivative at the endpoints of the interval and
subtracting.
Proof
Let P = {x;},i = 0, 1,..., n be a regular partition of [a, b] . Then, we can write
F (b) – F (a) = F (x„) – F (xo)
= [F (x„) – F (x„-1)] + [F (x,-1) – F (x„ 2)] + ...+ [F (x1) – F (xo)]
=2 F (z:) – F (z;-1) -
M
•A 3:04
![b The distribution of incon x
O Homework: Section 5.4
= 5.3 The Fundamental Th
b Answered: exercises, use x
O Selena Gomez - Slov
O Qun fine settimana - 17 x
+
A https://openstax.org/books/calculus-volume-1/pages/5-3-the-fundamental-theorem-of-calculus#fs-id1170572376509
< Calculus Volume 1
5.3 The Fundamental Theorem of Calculus
E Table of contents
Search this book
9 My highlights
B Print
= [F(x„) – F (xn-1)] + [F" (xn-1) – F (xn-2)] + ... + [F°(x1) – F'(xo)J
[F (x;) – F (x;-1)] .
Now, we know F is an antiderivative of f over [a, b] , so by the Mean Value Theorem (see The Mean Value Theorem) for
i = 0,1,..., n we can find c; in [x;_1, x;] such that
F (x;) – F (x;-1) = F'(c;) (x; – ¤;-1) = f (c;) Aæ.
Then, substituting into the previous equation, we have
F (b) – F (a) =
Taking the limit of both sides as n → 00, we obtain
F (b) – F (a)
= lim
f (c;) Ax
i=1
M
2 v A 3:01](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d6a2302-be00-405d-bbf6-4b5c194aa6b1%2F7a18b58e-008b-42e4-ab88-6a2723ac1213%2Fkg442h8_processed.png&w=3840&q=75)
Transcribed Image Text:b The distribution of incon x
O Homework: Section 5.4
= 5.3 The Fundamental Th
b Answered: exercises, use x
O Selena Gomez - Slov
O Qun fine settimana - 17 x
+
A https://openstax.org/books/calculus-volume-1/pages/5-3-the-fundamental-theorem-of-calculus#fs-id1170572376509
< Calculus Volume 1
5.3 The Fundamental Theorem of Calculus
E Table of contents
Search this book
9 My highlights
B Print
= [F(x„) – F (xn-1)] + [F" (xn-1) – F (xn-2)] + ... + [F°(x1) – F'(xo)J
[F (x;) – F (x;-1)] .
Now, we know F is an antiderivative of f over [a, b] , so by the Mean Value Theorem (see The Mean Value Theorem) for
i = 0,1,..., n we can find c; in [x;_1, x;] such that
F (x;) – F (x;-1) = F'(c;) (x; – ¤;-1) = f (c;) Aæ.
Then, substituting into the previous equation, we have
F (b) – F (a) =
Taking the limit of both sides as n → 00, we obtain
F (b) – F (a)
= lim
f (c;) Ax
i=1
M
2 v A 3:01
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