Setup the limit definition of definite integrals?

So you just need to set up the limit definition.

You would need to use the formula 
[tex]\displaystyle \int_{-1}^2(1+x^2)\mathrm dx = \lim_{n\to\infty}\sum_{i=1}^n f(x_i)\Delta x[/tex]

Where [tex]x_i = a + i\Delta x[/tex] and [tex]\Delta x = \dfrac{b-a}n[/tex].

Basically we would have approximate area with [tex]n[/tex] rectangles with width [tex]\Delta x[/tex] and height [tex]f(x_i)[/tex], and we have the value of n approaches to infinity.

Now, from the definite integral, we have [tex]a = -1, b = 2, f(x) = 1 + x^2[/tex].

So then that would be
[tex]\displaystyle \int_{-1}^2(1+x^2)\mathrm dx = \lim_{n\to\infty}\sum_{i=1}^n f(x_i)\Delta x \\= \lim_{n\to\infty}\sum_{i=1}^n (1+x_i^2)\dfrac{2 --1}n \\= \lim_{n\to\infty}\sum_{i=1}^n \left(1+\left(-1 + \dfrac3ni\right)^2\right)\dfrac{3}n[/tex]

So this is how you set up the limit definition of definite integrals.

Hope this helps.