In this problem you will calculate the area between the curve y = f(x) = 7x² +8 and the x-axis over the interval [0, 4] using a limit of right-endpoint Riemann sums: You may use the fact that Ik = Area = lim d. Find f(x) in terms of k and n. f(xk) = 2 e. Find f(x)Ax in terms of k and n. f(xk)Ax = n-x k=1 a. Start by subdividing [0, 4] into n subintervals [0, ₁],[₁, 2], ..., [Tn-1, En], all of the same width, A. Express the subinterval width Ax in terms of n. Ax= 4/n k=1 b. The first, second, and third subintervals in the scheme above are [0, £1], [X1, X2], [#2, #3]. Find their right endpoints #1, #2, #3. Enter your answers, which should depend on n, as a comma-separated list. #1, #2, I3 = f(xx) Ax Σ₁² = n(n+1) (2n + 1). i=1 c. The kth subinterval in the scheme above (assuming 1 ≤ k ≤ n) is [æk-1, *k]. Find a formula involving k and n for its right endpoint: (1) f. Find the value of the right-endpoint Riemann sum in terms of n. f(xk)Ax = g. Find the limit of the right-endpoint Riemann sums. This completes the area-finding project described in line (1) above. Time (21(2.) 4x) -
In this problem you will calculate the area between the curve y = f(x) = 7x² +8 and the x-axis over the interval [0, 4] using a limit of right-endpoint Riemann sums: You may use the fact that Ik = Area = lim d. Find f(x) in terms of k and n. f(xk) = 2 e. Find f(x)Ax in terms of k and n. f(xk)Ax = n-x k=1 a. Start by subdividing [0, 4] into n subintervals [0, ₁],[₁, 2], ..., [Tn-1, En], all of the same width, A. Express the subinterval width Ax in terms of n. Ax= 4/n k=1 b. The first, second, and third subintervals in the scheme above are [0, £1], [X1, X2], [#2, #3]. Find their right endpoints #1, #2, #3. Enter your answers, which should depend on n, as a comma-separated list. #1, #2, I3 = f(xx) Ax Σ₁² = n(n+1) (2n + 1). i=1 c. The kth subinterval in the scheme above (assuming 1 ≤ k ≤ n) is [æk-1, *k]. Find a formula involving k and n for its right endpoint: (1) f. Find the value of the right-endpoint Riemann sum in terms of n. f(xk)Ax = g. Find the limit of the right-endpoint Riemann sums. This completes the area-finding project described in line (1) above. Time (21(2.) 4x) -
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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![In this problem you will calculate the area between the curve y = f(x) = 7x² + 8 and the x-axis over the interval [0, 4] using a limit of right-endpoint Riemann
sums:
You may use the fact that
1, 2, 3 =
d. Find f(x) in terms of k and n..
f(xk) =
Area = lim
e. Find f(x) Ax in terms of k and n.
f(xk)Ax =
2.
n→∞0
k=1
k=1
a. Start by subdividing [0, 4] into n subintervals [To, 1], [₁, 2], ..., [n-1, n], all of the same width, Az. Express the subinterval width Ax in terms of n.
Ax= 4/n
i=1
b. The first, second, and third subintervals in the scheme above are [0, ₁], [₁, 2], [2, 3]. Find their right endpoints #1, #2, #3. Enter your answers, which
should depend on n, as a comma-separated list.
k=1
f(xk) Ax
c. The 4th subinterval in the scheme above (assuming 1 ≤ k ≤n) is [k-1, k]. Find a formula involving k and n for its right endpoint:
x = |
f. Find the value of the right-endpoint Riemann sum in terms of n.
f(x₁) Ax =
₁² = ²√n (n + 1) (2n + 1).
6
(1)
g. Find the limit of the right-endpoint Riemann sums. This completes the area-finding project described in line (1) above.
lim
f(xk) Ax| =
TL-700](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3f4a3967-3003-4c89-858c-e494ea95ddce%2F88da09b6-06a7-48d6-92f7-068d5939d303%2F385nxxa_processed.png&w=3840&q=75)
Transcribed Image Text:In this problem you will calculate the area between the curve y = f(x) = 7x² + 8 and the x-axis over the interval [0, 4] using a limit of right-endpoint Riemann
sums:
You may use the fact that
1, 2, 3 =
d. Find f(x) in terms of k and n..
f(xk) =
Area = lim
e. Find f(x) Ax in terms of k and n.
f(xk)Ax =
2.
n→∞0
k=1
k=1
a. Start by subdividing [0, 4] into n subintervals [To, 1], [₁, 2], ..., [n-1, n], all of the same width, Az. Express the subinterval width Ax in terms of n.
Ax= 4/n
i=1
b. The first, second, and third subintervals in the scheme above are [0, ₁], [₁, 2], [2, 3]. Find their right endpoints #1, #2, #3. Enter your answers, which
should depend on n, as a comma-separated list.
k=1
f(xk) Ax
c. The 4th subinterval in the scheme above (assuming 1 ≤ k ≤n) is [k-1, k]. Find a formula involving k and n for its right endpoint:
x = |
f. Find the value of the right-endpoint Riemann sum in terms of n.
f(x₁) Ax =
₁² = ²√n (n + 1) (2n + 1).
6
(1)
g. Find the limit of the right-endpoint Riemann sums. This completes the area-finding project described in line (1) above.
lim
f(xk) Ax| =
TL-700
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