Consider the function f(x)=x³ on the interval [0,1]. (a) Suppose we divide the interval [0,1] into n equal subintervals, and use left-endpoints to construct a Riemann sum for the function f. Use the sigma notation to write an expression for this Riemann sum. You do not need to simplify it. Note that your answer should be explicit and depend on n only. It should not include symbols such as f,Ax or ck. (b) Is the Riemann sum obtained in part (a) an overestimate or underestimate of the area between the graph of the function and the x-axis for 0
Consider the function f(x)=x³ on the interval [0,1]. (a) Suppose we divide the interval [0,1] into n equal subintervals, and use left-endpoints to construct a Riemann sum for the function f. Use the sigma notation to write an expression for this Riemann sum. You do not need to simplify it. Note that your answer should be explicit and depend on n only. It should not include symbols such as f,Ax or ck. (b) Is the Riemann sum obtained in part (a) an overestimate or underestimate of the area between the graph of the function and the x-axis for 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Consider the function f(x)=x on the interval [0,1].
(a) Suppose we divide the interval [0,1] into n equal subintervals, and use left-endpoints to construct
a Riemann sum for the function f.
Use the sigma notation to write an expression for this Riemann sum. You do not need to simplify it.
Note that your answer should be explicit and depend on n only. It should not include symbols such as
f, Дх or ck.
(b) Is the Riemann sum obtained in part (a) an overestimate or underestimate of the area between the
graph of the function and the x-axis for 0<x<1? Explain.
(Answer the question without calculating the exact area.)
(c) The Riemann sum obtained in part (a) can be simplified to the following expression:
(п — 1)? . (2n2 — 2n - 1)
12n4
1
Using this simplified expression, calculate the definite integral
x' dx.
Show your work and do not use antiderivatives or the Fundamental Theorem of Calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9df1e594-4936-43e0-b324-bcf87eae80a9%2F7d684a36-aca4-40e5-a707-8f805c5626c4%2F0fu9xn_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the function f(x)=x on the interval [0,1].
(a) Suppose we divide the interval [0,1] into n equal subintervals, and use left-endpoints to construct
a Riemann sum for the function f.
Use the sigma notation to write an expression for this Riemann sum. You do not need to simplify it.
Note that your answer should be explicit and depend on n only. It should not include symbols such as
f, Дх or ck.
(b) Is the Riemann sum obtained in part (a) an overestimate or underestimate of the area between the
graph of the function and the x-axis for 0<x<1? Explain.
(Answer the question without calculating the exact area.)
(c) The Riemann sum obtained in part (a) can be simplified to the following expression:
(п — 1)? . (2n2 — 2n - 1)
12n4
1
Using this simplified expression, calculate the definite integral
x' dx.
Show your work and do not use antiderivatives or the Fundamental Theorem of Calculus.
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