Question

Use the formulas for power sums to approximate the area under the graph of over the interval [1,3]. f(x)=2x^2+17.Compute using a left‑endpoint approximation.A=limN→[infinity] L (subscript) NEnter an exact answer. Do not round.A=?

Answer 1

The area under the graph of the function f(x) = 2x^2 + 17 from x=1 to x=3 using a left-endpoint approximation can be found by computing the limit of the Riemann sum as N approaches infinity, representing the definite integral of the function over the interval.

Step-by-step explanation:

To approximate the area under the graph of the function f(x) = 2x2 + 17 over the interval [1,3] using a left-endpoint approximation, we can use the Riemann sum approach considering the limit as the number of subintervals, N, approaches infinity. The left-endpoint approximation is given by:

LN = Σi=1N f(xi-1)Δx

Where Δx = (b - a)/N and xi-1 = a + (i - 1)Δx for a left endpoint. For the function f(x) and the interval [1,3], Δx = (3 - 1)/N and xi-1 = 1 + (i - 1)(2/N). Substituting the function into the sum and taking the limit as N approaches infinity gives us the exact area A under the curve:

A = limN→∞ (Σi=1N (2(1 + (i - 1)(2/N))2 + 17)(2/N))

This limit represents the definite integral of f(x) from x=1 to x=3, which can be computed to find the exact area A.