Question

Express the integral as a limit of Riemann sums using right endpoints. Do not evaluate the limit.

a) ∫[4 to 6] x² dx
b) ∫[6 to 4] x² dx
c) ∫[4 to 6] x² dx
d) ∫[6 to 4] x² dx

Answer 1

To express the integral as a limit of Riemann sums using right endpoints, divide the interval into subintervals and use the Riemann sum formula.

Step-by-step explanation:

To express the integral as a limit of Riemann sums using right endpoints, we need to partition the interval [a, b] into subintervals of equal width. Let's consider the integral ∫[4 to 6] x² dx as an example:

We can divide the interval [4, 6] into n subintervals of width Δx = (b - a)/n. The right endpoints of these subintervals will be x_i = a + iΔx, where i = 1, 2, ..., n. The Riemann sum for this integral is:

∑(i=1 to n) f(x_i)Δx

where f(x) = x². Similarly, we can express the other integrals ∫[6 to 4] x² dx, ∫[4 to 6] x² dx, and ∫[6 to 4] x² dx as limits of Riemann sums using right endpoints.