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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(2) dr = lim U, n00 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, and L, 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: 12 +2° + 32 + 4? + 5² + ...+ (n – 1)² + n² = k? = n(n+1)(2n + 1) 6 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 U, 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.

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(2) dr = lim U, n00 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, and L, 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: 12 +2° + 32 + 4? + 5² + ...+ (n – 1)² + n² = k? = n(n+1)(2n + 1) 6 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 U, 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.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.5: Applications
Problem 28EQ
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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 a lim Ln E| f(x) dx = lim U, n00 n+00 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)? + n² = > k² - 6 k=1 (n + 1)(2n + 1) (п — 1)(2п — 1) Use this fact to show that Un and (harder) that Ln = 6n2 6n2 (d) Find lim Un and lim Ln. n00 Verify, using the Fundamental Theorem of Calcu- n00 lus (that is, conventional methods of integration), that these limits are equal to 1 x² dx.
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 a lim Ln E| f(x) dx = lim U, n00 n+00 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)? + n² = > k² - 6 k=1 (n + 1)(2n + 1) (п — 1)(2п — 1) Use this fact to show that Un and (harder) that Ln = 6n2 6n2 (d) Find lim Un and lim Ln. n00 Verify, using the Fundamental Theorem of Calcu- n00 lus (that is, conventional methods of integration), that these limits are equal to 1 x² dx.
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