Using 4 equal-width intervals, show that the trapezoidal rule is the average of the upper and lower sum estimates for the integral from 0 to 2 of x squared, dx .

Answer 1

a few seconds with google will turn up several good online calculators for this and similar problems. You can use them to verify your results.

Answer 2

What does it mean by "upper and lower estimates?"
How do I find these?

Answer 3

Of the left and right sums, one will overestimate and the other will underestimate.

Since the area of a trapezoid (with vertical bases h1 and h2) is (h1+h2)*w
You can see how that is the average of the left-side height and the right-side height for each sub-interval.

I can't believe this is not explained in your text, and in far greater detail, giving illustrations!