f. Find the value of the right-endpoint Riemann sum in terms of n. ΣΗ(2) Δα k=1 g. Find the limit of the right-endpoint Riemann sum. lim (Σ f(xk)Δα f(en)Δ.) η 100 k=1 =
f. Find the value of the right-endpoint Riemann sum in terms of n. ΣΗ(2) Δα k=1 g. Find the limit of the right-endpoint Riemann sum. lim (Σ f(xk)Δα f(en)Δ.) η 100 k=1 =
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
F&G
![In this problem, you will calculate the area between \( f(x) = x^2 \) and the x-axis over the interval \([3, 10]\) using a limit of right-endpoint Riemann sums:
\[ \text{Area} = \lim_{{n \to \infty}} \left( \sum_{{k=1}}^{n} f(x_k) \Delta x \right). \]
Express the following quantities in terms of \( n \), the number of rectangles in the Riemann sum, and \( k \), the index for the rectangles in the Riemann sum.
a. We start by subdividing \([3, 10]\) into \( n \) equal width subintervals \([x_0, x_1], [x_1, x_2], \ldots, [x_{n-1}, x_n]\) each of width \(\Delta x\). Express the width of each subinterval \(\Delta x\) in terms of the number of subintervals \( n \).
\[ \Delta x = \frac{(10-3)}{n} \]
b. Find the right endpoints \( x_1, x_2, x_3 \) of the first, second, and third subintervals \([x_0, x_1], [x_1, x_2], [x_2, x_3]\) and express your answers in terms of \( n \).
\[ x_1, x_2, x_3 = \frac{(3n+7)}{n}, \frac{(3n+14)}{n}, \frac{(3n+21)}{n} \]
(Enter a comma separated list.)
c. Find a general expression for the right endpoint \( x_k \) of the \( k^{th} \) subinterval \([x_{k-1}, x_k]\) where \( 1 \leq k \leq n \). Express your answer in terms of \( k \) and \( n \).
\[ x_k = \frac{(3n+7k)}{n} \]
d. Find \( f(x_k) \) in terms of \( k \) and \( n \).
\[ f(x_k) = \left[ \frac{(3n+7k)}{n](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97f77314-22cd-4899-bd5f-be3ef9b0bc2e%2Fdbdc8ac5-ad72-45b8-a2a7-955d23c1d61a%2Fyz5zbrr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:In this problem, you will calculate the area between \( f(x) = x^2 \) and the x-axis over the interval \([3, 10]\) using a limit of right-endpoint Riemann sums:
\[ \text{Area} = \lim_{{n \to \infty}} \left( \sum_{{k=1}}^{n} f(x_k) \Delta x \right). \]
Express the following quantities in terms of \( n \), the number of rectangles in the Riemann sum, and \( k \), the index for the rectangles in the Riemann sum.
a. We start by subdividing \([3, 10]\) into \( n \) equal width subintervals \([x_0, x_1], [x_1, x_2], \ldots, [x_{n-1}, x_n]\) each of width \(\Delta x\). Express the width of each subinterval \(\Delta x\) in terms of the number of subintervals \( n \).
\[ \Delta x = \frac{(10-3)}{n} \]
b. Find the right endpoints \( x_1, x_2, x_3 \) of the first, second, and third subintervals \([x_0, x_1], [x_1, x_2], [x_2, x_3]\) and express your answers in terms of \( n \).
\[ x_1, x_2, x_3 = \frac{(3n+7)}{n}, \frac{(3n+14)}{n}, \frac{(3n+21)}{n} \]
(Enter a comma separated list.)
c. Find a general expression for the right endpoint \( x_k \) of the \( k^{th} \) subinterval \([x_{k-1}, x_k]\) where \( 1 \leq k \leq n \). Express your answer in terms of \( k \) and \( n \).
\[ x_k = \frac{(3n+7k)}{n} \]
d. Find \( f(x_k) \) in terms of \( k \) and \( n \).
\[ f(x_k) = \left[ \frac{(3n+7k)}{n
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
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,
