Question

Write the integral of the region as the limit of a Riemann sum with intervals of equal width and right endpoints as sample points.

3
∫ (x³ - 4x) dx
0

n
lim ∑ ............
n → [infinity] i = 1

Answer 1

To express the integral of the function f(x) = x^3 - 4x from 0 to 3 as a limit of a Riemann sum, we partition the interval [0,3] into n equal-width parts and as n approaches infinity, the Riemann sum converges to the value of the integral.

Step-by-step explanation:

To express the integral of the function f(x) = x³ - 4x from 0 to 3 using a Riemann sum, we first define the interval [0,3] to be subdivided into n equal-width partitions. The width of each subinterval, Δx, is ³/n. Letting xi denote the right endpoint of the i-th subinterval, we have xi = (i³)/n. The Riemann sum can then be represented as:

Σi=1^n f(xi)Δx

Substituting in the expression for f(xi) and Δx, we get:

Σi=1^n [(i³/n)³ - 4(i³/n)](³/n).

As the number of subintervals n goes to infinity, the Riemann sum approaches the exact value of the integral:

∑ n → ∞ Σi=1^n [(i³/n)³ - 4(i³/n)](³/n)