Final answer:
The expression for the area under the curve f(x) = 7x / (x² + 9) as a limit using right endpoints is the limit as n approaches infinity of Σ from i = 1 to n [(7(1 + i(2/n)))/((1 + i(2/n))² + 9) * (2/n)].
Step-by-step explanation:
When using a right endpoint to approximate the area under a curve, we divide the interval from 1 to 3 into n equal subintervals, each with a width of Δx = (3 - 1)/n = 2/n.
For the function f(x) = 7x / (x² + 9), the height at the right endpoint of each subinterval will be f(xi) where xi = 1 + i(2/n) for i = 1, 2, ..., n.
The limit of the sum of the areas of the rectangles as n approaches infinity is what defines the definite integral from 1 to 3 of f(x).
The expression for the area as a limit is:
Limit as n approaches infinity of Σ (from i = 1 to n) [f(xi) * Δx]
Substituting f(xi) and Δx gives:
Limit as n approaches infinity of Σ (from i = 1 to n) [(7xi)/(xi² + 9) * (2/n)]
Finally, replacing xi with 1 + i(2/n), we get:
Limit as n approaches infinity of Σ (from i = 1 to n) [(7(1 + i(2/n)))/((1 + i(2/n))² + 9) * (2/n)]