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 | f(x) dx = lim Un lim Ln = a 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 n(n + 1)(2n + 1) 12 + 22 + 32 + 4² + 5² + . + (n – 1)² + n² = k² = ... k=1 (n + 1)(2n + 1) (n – 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- n-00 lus (that is, conventional methods of integration), that these limits are equal to x² dx.
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 | f(x) dx = lim Un lim Ln = a 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 n(n + 1)(2n + 1) 12 + 22 + 32 + 4² + 5² + . + (n – 1)² + n² = k² = ... k=1 (n + 1)(2n + 1) (n – 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- n-00 lus (that is, conventional methods of integration), that these limits are equal to 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
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-
Un and Ln on n equal subintervals for the integral
mann sunms,
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
12 + 22 + 32 + 4? +52 +
+ (n – 1)² + n² =5k² = n(n+1)(2n + 1)
%3D
...
k=1
(n + 1)(2n + 1)
(n – 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
x² dx.
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