Problem 7 (b) Estimate the area under the graph of f using fourapproximating rectangles and taking the sample pointsto be (i) right endpoints and (ii) midpoints. In each casesketch the curve and the rectangles.(c) Improve your estimates in part (b) by using eightrectangles.7. Evaluate the upper and lower sums for f(x) = 2 + sin x,0 NOTE It can be shown that an equivalent definition of area is the followiunique number that is smaller than all the upper sums and bigger than all theWe saw in Examples 1 and 2, for instance, that the area (A = 5) is trappall the left approximating sums L, and all the right approximating sums R,.in those examples, f(x) = x², happens to be increasing on [0, 1] and so thearise from left endpoints and the upper sums from right endpoints. (See FigIn general, we form lower (and upper) sums by choosing the sample poif(x) is the minimum (and maximum) value of f on the ith subinterval. (and Exercises 7-8.)%3DУАFIGURE 14Lower sums (short rectangles) andupper sums (tall rectangles)b.

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Description: Problem 7 (b) Estimate the area under the graph of f using fourapproximating rectangles and taking the sample pointsto be (i) right endpoints and (ii) midpoints. In each casesketch the curve and the rectangles.(c) Improve your estimates in part (b) b

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(b) Estimate the area under the graph of f using four approximating rectangles and taking the sample points to be (i) right endpoints and (ii) midpoints. In each case sketch the curve and the rectangles. (c) Improve your estimates in part (b) by using eight rectangles. 7. Evaluate the upper and lower sums for f(x) = 2 + sin x, 0

(b) Estimate the area under the graph of f using four approximating rectangles and taking the sample points to be (i) right endpoints and (ii) midpoints. In each case sketch the curve and the rectangles. (c) Improve your estimates in part (b) by using eight rectangles. 7. Evaluate the upper and lower sums for f(x) = 2 + sin x, 0

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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Concept explainers

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.

Topic Video

Question

Problem 7

(b) Estimate the area under the graph of f using four
approximating rectangles and taking the sample points
to be (i) right endpoints and (ii) midpoints. In each case
sketch the curve and the rectangles.
(c) Improve your estimates in part (b) by using eight
rectangles.
7. Evaluate the upper and lower sums for f(x) = 2 + sin x,
0<x< T, with n = 2, 4, and 8. Illustrate with diagrams
like Figure 14.
8. Evaluate the upper and lower sums for f(x) = 1 + x².
-1 <x < 1, with n = 3 and 4. Illustrate with diagrams like
%3D
%3D
Figure 14.

NOTE It can be shown that an equivalent definition of area is the followi
unique number that is smaller than all the upper sums and bigger than all the
We saw in Examples 1 and 2, for instance, that the area (A = 5) is trapp
all the left approximating sums L, and all the right approximating sums R,.
in those examples, f(x) = x², happens to be increasing on [0, 1] and so the
arise from left endpoints and the upper sums from right endpoints. (See Fig
In general, we form lower (and upper) sums by choosing the sample poi
f(x) is the minimum (and maximum) value of f on the ith subinterval. (
and Exercises 7-8.)
%3D
УА
FIGURE 14
Lower sums (short rectangles) and
upper sums (tall rectangles)
b.

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