This lesson on approximating the area under the graph of a function is intended for students enrolled in AP Calculus AB or BC, Calculus Honors, or College Calculus.
Given a function f(x) where f(x)≥0 over an interval a ≤ x ≤ b, we investigate how to find the area of the region that is under the graph of f(x) and above x-axis over the interval [a,b]. In this topic, we will use formula
1 + 2 + 3 + ...+ n = n(n+1)/2, and
1^2 + 2^2 + 3^2 + 4^2 + ...+(n-1)^2 + n^2 = n (n+1)(2n+1)/6 to simply the sums.
Finally, we will take the limit of our area as n tends to infinity so that we can evaluate the exact area under the graph.
An example that can be generalized to any problem is given. AP Calculus students should be able to follow this example and duplicate this idea in any other related problem. The guidelines that are required for any other problem is given.
The other sum formula that is worthy to remember is:
1^3 + 2^3 + 3^3 + 4^3 + ...+ (n-1)^3 + n^3 = [ n(n+1)/2 ]^2