(3 points) In this problem you will calculate the area between f(x) = ² and the z-axis over the
interval [2, 9] using a limit of right-endpoint Riemann sums:
Area = lim (Σ.
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.
ΣH(24)Az).
k 1
a. We start by subdividing [2, 9] into n equal width subintervals
[0, 1],[1, 2],..., [Tn-1, En] each of width Az. Express the width of each subinterval
Az in terms of the number of subintervals n.
Ax= 7/n
b. Find the right endpoints ₁, 2, 3 of the first, second, and third subintervals
[0, 1],[1, ₂], [2, 3] and express your answers in terms of n
T1, T2, T3 2+7/n,2+14/n,2+21/n
d. Find f(x) in terms of k and n.
f(xk)
E (2+7k/n)^2
c. Find a general expression for the right endpoint of the kth subinterval [k-1, kl. where
1 kn. Express your answer in terms of k and n.
k
2+7k/n
e. Find f(x)Az in terms of k and n.
f(x) Ax= (2+7k/n)^2(7/n)
f. Find the value of the right-endpoint Riemann sum in terms of n.
TL
f(xx) Ax=
(4n+(28/n)(n(n+1)/2)+(49/n^2)(n(n+1)(2n+1)/6)(7/n))
lim
TL 200
(Enter a comma separated list.)
g. Find the limit of the right-endpoint Riemann sum.
(1)
f(xk) Δε ===240.33