169. ∫ (4 / x^2) dx over [1,4]
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
100%
In the following exercises, use a calculator to estimate the area under the curve by computing T10, the average of the left- and right-endpoint Riemann sums using N=10 rectangles. Then, using the Fundamental Theorem of Calculus, Part 2, determine the exact area.
165.
y = x^3 + 6x^2 + x − 5 over [−4,2]
169.
∫ (4 / x^2) dx over [1,4]
![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%2Fc7342d1b-4b2b-4c3d-a075-e500a5206c68%2F59t6l4e_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%2Fc7342d1b-4b2b-4c3d-a075-e500a5206c68%2Fzno3ced_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 2 steps with 1 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
