Use the Definition to find an expression for the area under the graph of f as a limit. Do not evaluate the limit.f(x) = x2 + 1 + 2x,    2 ≤ x ≤ 4lim n → ∞ ni = 1

The area under the curve of a function can be found using the definition of a definite integral as a limit of Riemann sums. Here's how you can express the area under the graph of f(x) = x^2 + 1 + 2x, from x = 2 to x = 4, as a limit:

1. Divide the interval [2, 4] into n equal subintervals, each of width Δx = (4 - 2)/n = 2/n.

2. Choose a sample point x_i* in each subinterval [x_i-1, x_i]. For simplicity, let's choose the right endpoint, so x_i* = x_i = 2 + i*(2/n).

3. Evaluate the function at each sample point to get the height of the rectangle in the i-th subinterval: f(x_i*) = (2 + i*(2/n))^2 + 1 + 2*(2 + i*(2/n)).

4. Multiply the height by the width to get the area of the rectangle in the i-th subinterval: ΔA_i = f(x_i*)Δx = [(2 + i(2/n))^2 + 1 + 2*(2 + i*(2/n))]*(2/n).

5. Sum the areas of all rectangles to approximate the total area under the curve: A ≈ Σ_{i=1}^n ΔA_i.

6. Take the limit as n approaches infinity to get the exact area under the curve: A = lim_{n→∞} Σ_{i=1}^n ΔA_i.

So, the expression for the area under the graph of f as a limit is:

A = lim_{n→∞} Σ_{i=1}^n [(2 + i*(2/n))^2 + 1 + 2*(2 + i*(2/n))]*(2/n).