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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
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.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
