Q:
Your objective is to find the integral correct.
Now i happen to find doing things like these and Riemann sums and the Simpson's and Midpoint rules hopelessly annoying. I have somewhat purposed in my heart to forget how to do them, so what i say may not be 100% correct, but you can verify what i say with your text.
$\displaystyle \int_0^b f(x) dx $
$\displaystyle = \int_0^3 f(x) dx $
$\displaystyle = \int_0^3 (1 + x)^2 dx $
$\displaystyle = \lim_{n \rightarrow \infty} [f(x_{1}) \triangle x + f(x_{2}) \triangle x + ... + f(x_{n}) \triangle x] $
$\displaystyle = \lim_{n \rightarrow \infty} \sum_{i = 1}^{\infty} f(x_{i}) \triangle x$