I wanted to try to perform a Riemann Sum for the following integral, but I got stuck in the middle.
$$\int_{-1}^0 e^{-x^2}\ \text{d}x$$
So the interval is $[-1, 0]$, and I chose $\Delta x = \dfrac{1}{n}$. In this way $x_i = \dfrac{i}{n}$ and $$f(x_i) = e^{-(i/n)^2}$$
Here is where I got stuck because now I cannot manage this:
$$\dfrac{1}{n} \sum_{i = 1}^n e^{-(i/n)^2}$$
How can I proceed?
I thought of using Geometric Progression but I cannot understand how to.
This answers the amended question in the comments, namely, “how does one show this integral is positive?”.
It’s an integral of a strictly positive function over a nondegenerate interval. You can very simply show this is positive without a specific calculation (which isn’t really possible in this case anyway). Use $e^{-x^2}\ge1/e$ so note that any partial Riemann sum, using any partition, shall give a sum greater than $1/e\cdot(0-(-1))=1/e$. Therefore the limit of the sums along any partitions whose mesh vanishes - the Riemann integral, by definition - shall be a limit of a sequence of numbers all greater than $1/e$. The final limit must then be $\ge1/e>0$ by basic limit properties.