Page 88 - Calculus for the Clueless, Calc II

Another method is to approximate the region using parabolas on the top. We will isolate one of these regions.
As you will see, this method works only if n is an even number of intervals.

2
2
Let the parabola be given by y = f(x) = ax + bx + c: y at the left end is y L = f(-h) = ah - bh + c; y in the middle
2
is y M = f(0) = c; and y at the right is y R = f(h) = ah + bh + c.
The area of this region is

Do something weird. Factor out h/3 because it works!!

Just like before, the lower base of the first region is the upper base of the second region. Four times the middle
never changes. The formula issss ... A = h/3(y 0 + 4y 1 + 2y 2 + 4y 3 + ... + 4y n - 1 + y n), n being even.

Example 3—

Let's do the same example. Approximate dx

As some of you know and the rest will find out soon, the approximate answer is that In 4 = 1.386294361.

Example 4—

-x
A. Graph y = xe . B. Find the area x>0. C. Find the volume if the region is rotated about the x-axis.
Graphing exponentials is new, but we'll review graphing in general, areas, volumes of revolution, improper
integrals, L'Hopital's rule, and integration by parts.