Question

College calculus. I've tried it a couple times but i can't seem to get the right answer

Answer 1

The area is the sum of 'n' rectangle areas.
The width of the rectangle is size of interval, (domain size)/n
The height of each rectangle is f(interval) as interval moves along domain.

For this example, domain size = 3-1 = 2
size of interval = 2/n
height varies from f(1) to f(3), increasing by 2/n each time.
f(1+(2/n)i)

Putting this together, the area is the sum of:
$\frac{2}{n}*f(1+\frac{2i}{n})$

Since you are given the function f(x). Sub input into f(x) to get area in terms of n and i.

$f(1+\frac{2i}{n}) = \frac{3(1+\frac{2i}{n})}{(1+\frac{2i}{n})^2 +8} \\ \\ =\frac{3n+6i}{n}*\frac{n^2}{(n+2i)^2 +8n^2} \\ \\ =\frac{3n^2 +6ni}{9n^2+4ni+4i^2}$

Finally, the summation is:
$\frac{2}{n}*\frac{3n^2 +6ni}{9n^2+4ni+4i^2} = \frac{6n +12i}{9n^2+4ni+4i^2} \\ \\ A =\lim_{n \to \infty} \sum_{i=1}^n \frac{6n +12i}{9n^2+4ni+4i^2}$