5.2 definite integrals. subintervals are often denoted by x because they represent the change in x...
TRANSCRIPT
5.2 Definite Integrals
Subintervals are often denoted by x because they represent the change in x…but you all know this at least from chemistry class, right?
When we find the area under a curve by adding rectangles, the answer is called a Riemann sum.
subinterval
interval
The width of a rectangle is called a subinterval.
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Now let’s do some more notation so that you will understand it when you see it in the text book…
When we find the area under a curve by adding rectangles, the answer is called a Riemann sum.
subinterval
interval
The width of a rectangle is called a subinterval.
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xxfn
k
k 1
)(
If we take n rectangles (subintervals) and add them all up, the summation would look like this:
rectangle height
rectangle base
Summation of n rectangles
Subintervals do not all have to be the same size.
Equal subintervals make for easier and faster calculation, but some curves call for rectangles of different bases depending upon the shape of the curve. While we won’t be doing any in this lesson, we should at least consider that possibility here.
When we find the area under a curve by adding rectangles, the answer is called a Riemann sum.
subinterval
interval
The width of a rectangle is called a subinterval.
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8V t
In calculus texts, the partition x is also denoted by P. Let’s look at one summation expression that uses this notation…
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subinterval
interval
If the partition is denoted by P, then the length of the longest subinterval is called the norm of P and is denoted by .P
As gets smaller, the approximation for the area gets better.
P
,a bif P is a partition of the intervalk
n
k
kP
xxf
1
0)(lim
We can also think of the size of this partition in the
same way as we think of x
And now for some more terminology:
if P is a partition of the intervalk
n
k
kP
xxf
1
0)(lim
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subinterval
interval
Increase the number of subintervals
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subinterval
interval
Do you remember how we can improve this approximation?
,a b
subinterval
interval
As n gets bigger, P gets smaller
Since we know how to take limits…
DON’T WE?
We can then send the number of partitions to infinity which will send the base of each rectangle (the size of each partition) to…
00
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if P is a partition of the intervalk
n
k
kP
xxf
1
0)(lim ,a b
If we use subintervals of equal length, then the length of a
subinterval is:b a
xn
The sum can then be given by:
is called the Riemann Sum of
over .f ,a bk
n
k
kP
xxf
1
0)(lim
Leibnitz introduced a simpler notation for the definite integral:
Notice how as x 0, the change in x becomes dx.
1
limn b
k ank
f c x f x dx
The Definite Integral over the interval [a,b]
b
adxxf )(
IntegrationSymbol
lower limit of integration
upper limit of integration
integrandvariable of integration
So what do all of these symbols mean?
Don’t forget that we are still finding the area under the curve. For #8-28 in 5.2, if you make a graph of the problem, you can find the integral easily.
b
adxxf )(Summation Symbol
Rectangle Height
Rectangle Base
This happens to be #1 in 5.2 but we will do it to help you through the notation when you do the homework:
2
0
2 dxx
n
kkk
Pxc
1
2
0||||lim over the interval [0, 2]
Rectangle Height Rectangle Base
…which means just find the area under the curve y = x2 from 0 to 2