Question

How can you express the integral ∫⁸₄ x² dx as a limit of Riemann sums using right endpoints? Without evaluating the limit, determine the width of each subinterval in terms of n.

Answer 1

The width of each subinterval in the limit of Riemann sums for the integral ∨⁸₄ x² dx using right endpoints is Δx = 4/n. This represents dividing the interval [4, 8] into n equal parts.

Step-by-step explanation:

To express the integral ∨⁸₄ x² dx as a limit of Riemann sums using right endpoints, we divide the interval [4, 8] into n subintervals of equal width. The width of each subinterval, Δx, is thus the total length of the interval divided by the number of subintervals. Therefore, Δx = (8 - 4)/n or Δx = 4/n.

The Riemann sum using right endpoints for a function f(x) on [4, 8] with n subintervals is given by Sn = Σk=1nf(xk)Δx, where xk is the right endpoint of the k-th subinterval. As n approaches infinity, the Riemann sum becomes the definite integral of the function from 4 to 8.