Answer
$\lim\limits_{n \to \infty}\sum_{i=1}^{n}\sqrt{4+(1+\frac{2i}{n})^2}~\cdot \frac{2}{n}$
Work Step by Step
$\Delta x = \frac{b-a}{n} = \frac{3-1}{n} = \frac{2}{n}$
$x_i^* = 1+\frac{2i}{n}$
We can express the integral as a limit of Riemann sums:
$\int_{a}^{b}f(x)~dx = \lim\limits_{n \to \infty}\sum_{i=1}^{n}f(x_i^*)\Delta x$
$\int_{1}^{3}\sqrt{4+x^2})~dx = \lim\limits_{n \to \infty}\sum_{i=1}^{n}\sqrt{4+(1+\frac{2i}{n})^2}~\cdot \frac{2}{n}$
