Find area under $y= x^2 - x^4$ from x=-1 to x=0 using the Riemann sum

Question

I'm trying to find the area under $y= x^2 - x^4$ from $x=-1$ to $x=0$ using the Riemann sum.

This is what I've done so far:

$\Delta x = 1/n$

$x_i =-1 + i/n$

$A = R_n = \lim_{n\to \infty} {\sum_{i=1}^{n} \Delta x ((x_i)^2 - (x_i)^4) }$

$A = \lim_{n\to \infty} \Delta x{\sum_{i=1}^{n} (-1 + i/n)^2 - (-1 + i/n)^4 }$

I'm not sure how to proceed from here. I tried expanding the terms $(x_i)^2$ and $(x_i)^4$ but that didn't seem to help.

Thanks for your help!

Answer 1

Note that $$\sum_{i=1}^{n} (-1 + i/n)^2=\sum_{i=1}^{n} (\frac{i-n}n)^2\\ =\sum_{i=0}^{n-1} (\frac{i}n)^2\\=\frac 1{n^2}\sum_{i=0}^{n-1} (i^2)$$ To go from the first line to the second we took the terms in the reverse order. Now perform the sum. The same thing happens for the other part. When you consider the $\Delta x$ all the terms but the first in each will go away when you let $n \to \infty$