# Definite Integrals

Document Sample

```					   Definite Integrals

Riemann Sums and Trapezoidal
Rule
Why all this work? Why learn
integration?
We are now going to look to find the
numerical value of any curve f(x) from a to
b, bounded by the x axis. (Fig 1)

The connection to integration that will be
explained tomorrow. Right now we will
concentrate on some of the methods that
can be used to find the area under any
curve.
Method 1: Riemann Sums
(See Fig 2) If we partition a particular
function into intervals from x0 (a) to xn (b)
with evenly spaced subintervals (xk, xk+1,
xk+2…) we could find the area under the
curve IF we could find the area of all those
little subintervals. The more subintervals,
the more accurate the area, right?
Lets look at these as rectangles.
Riemann Sums (cont.)
(Fig. 3) If we choose a particular number, c,
in each subinterval, f(c)·Δx = the area of
the particular rectangle. If we add all
these rectangles, we’ll get a pretty close
approximation of the total area under the
curve.
Notice:
Arectangle = length·width = f(c)·Δx
Riemann Sums (cont.)
Whether we use the left side of the rectangle
or the right side we can see that the more
rectangles the more accurate the area.
Here’s another
n
Sn   f(c k )x k adds all the rectangles.
k 1
n
Area under curve = lim n   f(c k )x k
k 1
Trapezoidal Rule
(Fig.4 ) Instead of rectangles, we look at the
subintervals as trapezoids. Again, the
more trapezoids you put in, the more
accurate the estimation.

What is the formula for the area of a
trapezoid?
For just one trapezoid
1             1
A  h(b1  b2 )  h(y 0  y 1 )
2             2
If I add all the trapezoids (n of them)
1                1               1                   1
h(y 0  y 1 )  h(y 1  y 2 )  h(y 2  y 3 )  ... h(y n 1  y n )
2                2               2                   2
1
 h(y 0  2y 1  2y 2  2y 3  ...  2y n 1  y n )
2
ba
h
n
n=number of sub intervals
Will we do this all the time now?
Good grief NO!! This is a very rote method.
But as you see from the illustrations, the
more intervals you have the more
accurately you can find the area
Example
1.   Use trapezoidal rule to partition x 2 into 4
2


subintervals to estimate x 2dx
1

TONIGHT’S PROBLEM

Use trapezoidal rule to partition      into
x3 4
2


subintervals to estimate x 3 dx
0

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 5 posted: 9/11/2012 language: Unknown pages: 9