c) Now compute the numerical value of the definite integral by determining the limit of the summation from b) as n approaches infinity. n (2 + 6x) dx = lim f(x;)Ax = %3D 3 i=1
c) Now compute the numerical value of the definite integral by determining the limit of the summation from b) as n approaches infinity. n (2 + 6x) dx = lim f(x;)Ax = %3D 3 i=1
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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Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus.
Riemann Integral
Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.
Question

Transcribed Image Text:c) Now compute the numerical value of the definite integral by determining the limit of
the summation from b) as n approaches infinity.
n
(2+ 6x) dx
= lim )f(x;)Ax
i=1
![In this problem, we will compute the following integral:
(2 + 6x) dx
We will compute the definite integral using the equality
f(x) dx =
n
b – a
lim f(x;)Ax where Ax =
and xi
= a + iAx.
n
i=1
a) Imagine dividing the interval [3, 9] into n subintervals of equal width, where A is
the width of each of these subintervals, and x; is the right-endpoint of the ith
subinterval. Determine Ax and x;, where each is a function in n.
i) Ax =
ii) x; =
b) Using the answers you found above, compute f(x;)Ax for f (x) = 2 + 6x as
i=1
a summation. Note that your answer is a function of n.
>f(x;)Ax
i=1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdedc6ff1-e5e7-48e1-8e69-1d621b5c4e4e%2Ff5690382-13a7-4551-91b1-f5ec8c873593%2Ftos3hbo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:In this problem, we will compute the following integral:
(2 + 6x) dx
We will compute the definite integral using the equality
f(x) dx =
n
b – a
lim f(x;)Ax where Ax =
and xi
= a + iAx.
n
i=1
a) Imagine dividing the interval [3, 9] into n subintervals of equal width, where A is
the width of each of these subintervals, and x; is the right-endpoint of the ith
subinterval. Determine Ax and x;, where each is a function in n.
i) Ax =
ii) x; =
b) Using the answers you found above, compute f(x;)Ax for f (x) = 2 + 6x as
i=1
a summation. Note that your answer is a function of n.
>f(x;)Ax
i=1
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