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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 | 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.

Algebra for College Students
Algebra for College Students
10th Edition
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter9: Polynomial And Rational Functions
Section9.2: Remainder And Factor Theorems
Problem 53PS
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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 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.
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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