The interval [1, 9] is partitioned into n subintervals [xk-1, xk] for k = 1, ..., n, each of width Ax. Choose any such that ¤k-1 ≤ x ≤ xk. Let the function f be continuous over [1,9]. Do the following. ² [₁²₁ (a) State the limit definition of f(x) dx. (b) Estimate the integral in (a) if f(x) = x² using a Riemann sum with n = 4 subintervals of equal width and sample points x k for k= 1, 2, 3, 4. = (c) Sketch f(x) = x² and the rectangles whose area is the Reimann sum in (b). Use this sketch to explain why the sum in (b) overestimates the value of the integral in (a) when f(x) = x².
The interval [1, 9] is partitioned into n subintervals [xk-1, xk] for k = 1, ..., n, each of width Ax. Choose any such that ¤k-1 ≤ x ≤ xk. Let the function f be continuous over [1,9]. Do the following. ² [₁²₁ (a) State the limit definition of f(x) dx. (b) Estimate the integral in (a) if f(x) = x² using a Riemann sum with n = 4 subintervals of equal width and sample points x k for k= 1, 2, 3, 4. = (c) Sketch f(x) = x² and the rectangles whose area is the Reimann sum in (b). Use this sketch to explain why the sum in (b) overestimates the value of the integral in (a) when f(x) = x².
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter5: Rings, Integral Domains, And Fields
Section5.4: Ordered Integral Domains
Problem 7E: For an element x of an ordered integral domain D, the absolute value | x | is defined by | x |={...
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![The interval [1, 9] is partitioned into n subintervals [™k–1,¤k] for k : 1, ..., n, each of
width Ax. Choose any such that xk-1 ≤ x ≤ xk. Let the function f be
continuous over [1, 9]. Do the following.
9
(a) State the limit definition of
(b) Estimate the integral in (a) if f(x) = x² using a Riemann sum with n = 4
subintervals of equal width and sample points x = xk for k = 1, 2, 3, 4.
f(x) dx.
(c) Sketch f(x) = x² and the rectangles whose area is the Reimann sum in (b).
Use this sketch to explain why the sum in (b) overestimates the value of the
integral in (a) when f(x) = x².](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff42c59a5-908c-4bcd-9e36-90d243a82d72%2F4cfbcdc0-fafd-4233-9eae-bb8103c5bd11%2F70qaolm_processed.png&w=3840&q=75)
Transcribed Image Text:The interval [1, 9] is partitioned into n subintervals [™k–1,¤k] for k : 1, ..., n, each of
width Ax. Choose any such that xk-1 ≤ x ≤ xk. Let the function f be
continuous over [1, 9]. Do the following.
9
(a) State the limit definition of
(b) Estimate the integral in (a) if f(x) = x² using a Riemann sum with n = 4
subintervals of equal width and sample points x = xk for k = 1, 2, 3, 4.
f(x) dx.
(c) Sketch f(x) = x² and the rectangles whose area is the Reimann sum in (b).
Use this sketch to explain why the sum in (b) overestimates the value of the
integral in (a) when f(x) = x².
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