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.
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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![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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