This question asks you to find an integral using Riemann sums. Recall from lectures, that a definite integral f (x) dx can be defined as lim Ln f(x) dx = lim Un where Un and Ln are upper and lower Riemann sums respectively. (a) Sketch a diagram showing the rectangles corresponding to upper and lower Rie- mann sums, Un and Ln on n equal subintervals for the integral x² dx. Include the ith subinterval in your diagram. (b) Write down expressions for Un and Ln using sigma notation. (c) The sum of the squares of the first n positive integers is given by the following formula: n п(п + 1)(2n + 1) 12 + 22 + 32 + 4² + 52 + ...+ (n – 1)? + n² = > k² 6. k=1 (n + 1)(2n + 1) (п — 1)(2n — 1) Use this fact to show that Un and (harder) that Ln 6n2 6n2 (d) Find lim Un and lim Ln. Verify, using the Fundamental Theorem of Calcu- n00 lus (that is, conventional methods of integration), that these limits are equal to 1 x² dx.
This question asks you to find an integral using Riemann sums. Recall from lectures, that a definite integral f (x) dx can be defined as lim Ln f(x) dx = lim Un where Un and Ln are upper and lower Riemann sums respectively. (a) Sketch a diagram showing the rectangles corresponding to upper and lower Rie- mann sums, Un and Ln on n equal subintervals for the integral x² dx. Include the ith subinterval in your diagram. (b) Write down expressions for Un and Ln using sigma notation. (c) The sum of the squares of the first n positive integers is given by the following formula: n п(п + 1)(2n + 1) 12 + 22 + 32 + 4² + 52 + ...+ (n – 1)? + n² = > k² 6. k=1 (n + 1)(2n + 1) (п — 1)(2n — 1) Use this fact to show that Un and (harder) that Ln 6n2 6n2 (d) Find lim Un and lim Ln. Verify, using the Fundamental Theorem of Calcu- n00 lus (that is, conventional methods of integration), that these limits are equal to 1 x² dx.
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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Transcribed Image Text:4. This question asks you to find an integral using Riemann sums. Recall from lectures,
that a definite integral
f (x) dx can be defined as
9.
lim Ln
f (x) dx = lim Un
=
where Un and Ln are upper and lower Riemann sums respectively.
(a) Sketch a diagram showing the rectangles corresponding to upper and lower Rie-
mann sums, U,n and Ln on n equal subintervals for the integral
1
x² dx.
Include the ith subinterval in your diagram.
(b) Write down expressions for Un and Ln using sigma notation.
(c) The sum of the squares of the first n positive integers is given by the following
formula:
п
п(п + 1)(2n + 1)
12 + 22 + 32 + 4² + 5² +
+ (n – 1)² + n² =k?
k=1
(n + 1)(2n + 1)
(п — 1)(2n — 1)
Use this fact to show that U,n =
and (harder) that Ln
6n2
6n2
(d) Find lim Un and lim Ln. Verify, using the Fundamental Theorem of Calcu-
lus (that is, conventional methods of integration), that these limits are equal to
1
x² dx.
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