Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.7: Distinguishable Permutations And Combinations
Problem 30E
Related questions
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
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage