Approximate the area between the curve and the x-axis on the interval [0,2] using 4 rectangles and ...

(a) the Trapezoidal Rule

(b) Simpson's Rule (Note: Simpson's Rule is not on the AP exam, but it is in most college calculus courses)

Difficulty:

Solution:

(a) The Trapezoidal Rule for this function on [0,2] with n = 4 rectangles looks like:

The intial fraction is the difference of the endpoints of the interval divided by twice the number of trapezoids you're using. You then multiply by the function evaluated at each endpoint and at every appropriate step in between. Since you're using 4 trapezoids on an interval of length 2, those steps should each be 1/2. Add up all of those function values (multiplying each except for the endpoints) by 2, and you get your answer.

b) Simpson's Rule is very similar, except the initial fraction has a denominator of three times the number of intervals instead of 2 times it. In addition, you multiply the second function value by 4, the next by 2, then 4, then 2, then 4, etc. Just like the Trapezoidal Rule, the endpoints don't get multiplied by anything. A word of warning: Simpson's Rule only works when there are an even number of intervals.