Before viewing the following animations you might want
to see the computer lab on Riemann Sums on the Computer Lab Page first for a complete discussion of Riemann Sums. The first animation below show the left and right sums for f(x) = x^{2} on the
interval [1,2] The second animation shows the midpoint rule for the
same function and the same interval. Note since f(x)
= x^{2 }is an increasing function the right hand sum is an over estimate and the left hand sum is an under estimate. The exact answer is 7/3 0r 2.333. Note the midpoint rule converges much faster than
either the left hand or right hand rules.
Computer Lab Assignment page
Left and Right Sums
Midpoint Sum
The following 2 animations are for the function f(x) =
e ^{x} on [0,2]. The first animation are the left and right hand sums. this time since f(x) = e ^{x} is decreasing
the right hand sum is an under estimate and the left hand sum is is an over estimate. The second animation is the midpoint rule. Note the exact answer is .865.
Left and Right Sums
Midpoint Sum
Proof of the Fundamental Theorem of Calculus
The discussion of the trapezoid rule and Simpson's
Rule2 methods for approximating the definite integral are found on the Computer Lab Assignment Page.
