The interval [1, 9] is partitioned into n subintervals [xk-1, xk] 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. (a) State the limit definition of [₁² 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 = xk 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 xk-1 ≤ x ≤ xk. Let the function f be continuous over [1, 9]. Do the following. (a) State the limit definition of [₁² 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 = xk 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. =
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter3: The Derivative
Section3.3: Rates Of Change
Problem 25E
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![The interval [1, 9] is partitioned into n subintervals [xk-1, xk] for k = 1, ..., n, each of
width Ax. Choose any x such that xê-1 ≤ x* ≤ xk. Let the function f be
continuous over [1, 9]. Do the following.
[²³³
(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 = xk 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff3cf874b-7a7b-478f-a7b8-421442e72224%2Fa1ab0e70-30b9-459d-9f4f-49a9694f8192%2Fpk114aj_processed.png&w=3840&q=75)
Transcribed Image Text:The interval [1, 9] is partitioned into n subintervals [xk-1, xk] for k = 1, ..., n, each of
width Ax. Choose any x such that xê-1 ≤ x* ≤ xk. Let the function f be
continuous over [1, 9]. Do the following.
[²³³
(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 = xk 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.
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