In this problem you will calculate the area between f(x) = x² and the x-axis over the interval[2,9] using a limit of right-endpoint Riemann sums:nArea =lim Ef(x)Axk=1Express the following quantities in terms of n, the number of rectangles in the Riemann sum, and k,the index for the rectangles in the Riemann sum.a. We start by subdividing [2, 9] into n equal width subintervals [x0, x1], [x1, X2], ... , [Xp-1, Xn ]each of width Ax. Express the width of each subinterval Ar in terms of the number ofsubintervals n.Ax =7/nb. Find the right endpoints x1, X2, X3 of the first, second, and third subintervals[X0, X1 ], [x1, X2], [x2, x3] and express your answers in terms of n.X1, X2, X3 = 2+7/n, 2+14/n, 2+21/n(Enter a comma separated list.)c. Find a general expression for the right endpoint x of the kth subinterval [xx-1, Xx], where1

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In this problem you will calculate the area between f(x) = x² and the x-axis over the interval [2,9] using a limit of right-endpoint Riemann sums: n Area = lim Σf() Δx k=1 Express the following quantities in terms of n, the number of rectangles in the Riemann sum, and k, the index for the rectangles in the Riemann sum. a. We start by subdividing [2, 9] into n equal width subintervals [xo, x1], [x1, x2], ... , [Xp-1, Xn] each of width Ax. Express the width of each subinterval Ax in terms of the number of subintervalsn. Ax = 7/n b. Find the right endpoints x1, x2, X3 of the first, second, and third subintervals [X0, x1 ], [x1, x2], [x2, x3] and express your answers in terms of n. X1, X2, X3 = 2+7/n, 2+14/n, 2+21/n (Enter a comma separated list.) c. Find a general expression for the right endpoint xỵ of the kth subinterval [xx-1, Xg], where 1

In this problem you will calculate the area between f(x) = x² and the x-axis over the interval [2,9] using a limit of right-endpoint Riemann sums: n Area = lim Σf() Δx k=1 Express the following quantities in terms of n, the number of rectangles in the Riemann sum, and k, the index for the rectangles in the Riemann sum. a. We start by subdividing [2, 9] into n equal width subintervals [xo, x1], [x1, x2], ... , [Xp-1, Xn] each of width Ax. Express the width of each subinterval Ax in terms of the number of subintervalsn. Ax = 7/n b. Find the right endpoints x1, x2, X3 of the first, second, and third subintervals [X0, x1 ], [x1, x2], [x2, x3] and express your answers in terms of n. X1, X2, X3 = 2+7/n, 2+14/n, 2+21/n (Enter a comma separated list.) c. Find a general expression for the right endpoint xỵ of the kth subinterval [xx-1, Xg], where 1

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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Concept explainers

Riemann Sum

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

In this problem you will calculate the area between f(x) = x² and the x-axis over the interval
[2,9] using a limit of right-endpoint Riemann sums:
n
Area =
lim Ef(x)Ax
k=1
Express the following quantities in terms of n, the number of rectangles in the Riemann sum, and k,
the index for the rectangles in the Riemann sum.
a. We start by subdividing [2, 9] into n equal width subintervals [x0, x1], [x1, X2], ... , [Xp-1, Xn ]
each of width Ax. Express the width of each subinterval Ar in terms of the number of
subintervals n.
Ax =
7/n
b. Find the right endpoints x1, X2, X3 of the first, second, and third subintervals
[X0, X1 ], [x1, X2], [x2, x3] and express your answers in terms of n.
X1, X2, X3 = 2+7/n, 2+14/n, 2+21/n
(Enter a comma separated list.)
c. Find a general expression for the right endpoint x of the kth subinterval [xx-1, Xx], where
1 <k < n. Express your answer in terms of k and n.
Xk =
2+7k/n
d. Find f(xx) in terms of k and n.
f(xx) = | (2+7k/n)^2
e. Find f(xx)Ax in terms of k and n.
f(xx)Ax = (2+7k/n)^2(7/n)
f. Find the value of the right-endpoint Riemann sum in terms of n.
n
k=1
g. Find the limit of the right-endpoint Riemann sum.
n
lim
Ef(x)Ax ) =
k=1

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