area Integration
The problem
Integration
Denite integrals
The area problem
We know how to compute areas of triangles and polygons, but how do we compute areas of more general shapes? For example, how can we nd the area under the graph y = f (x) over an interval [a, b]?
The Fundamental Theorem of Calculus
Suppose that f is continuous on some interval I containing the point a. x
1
Let F (x) be dened for x in I by F (x) = dierentiable on I and F (x) = f (x), i.e. d dx
x a
a
f (t) dt. Then F is
Riemann sums
If f is continuous on [a, b], we can split the interval into N equal sub-intervals [xi1 , xi ], with xi xi1 = x = (b a)/N . We dene the mid-points x = xi1 +xi . Then we can dene the sums i 2
N 1 N N
f (t) dt = f (x).
2
If G is any antiderivative of f on I, so that G (x) = f (x) on I, then
b a
LN =
i=0
f (xi )x,
RN =
i=1
f (xi )x, and MN =
i=1
f (x )x. i
f (x) dx = G(b) G(a).
As N , these sums all tend to the same limit. We write this limit as
b a
f (x) dx.
Tony Ware, University of Calgary
Winter 2009: Calculus I: Week 12
5.1: 7-24
Tony Ware, University of Calgary
Winter 2009: Calculus I: Week 12
5.3: 1-52, 5.4: 1-23
Integration
The area problem
Integration
Denite integrals
Riemann sums - an example
What is the area under f (x) = x between x = 0 and x = 1?
We discover that, with N equal-width intervals, LN 1 1 1 1 = < area < + = RN . 2 2N 2 2N 1 . 2
Denite integrals
Recall that the FTC tells us that in order to evaluate I=
b a
f (x) dx,
So since this must be true for all N , the area is In this example we have
1 0
where f is continuous on [a, b], all we have to do is nd a function F (x) satisfying F (x) = f (x), and then I = F (b) F (a) = F (x)]a . If f is not continuous, we split [a, b] up into smaller intervals on which f is continuous.
b
x dx =
1 . 2
Denite integrals
An of expression the form
a b
f (x) dx
is known as a denite integral.
Tony Ware, University of Calgary Winter 2009: Calculus I: Week 12 5.1: 7-24 Tony Ware, University of Calgary Winter 2009: Calculus I: Week 12 5.3: 1-52, 5.4: 1-23
Integration
Denite integrals
Integration
Denite integrals
Some facts about denite integrals
They are just numbers, and the variable used does not matter:
b a
Integrals as functions
The rst part of the FTC tells us that if F (x) = a f (t) dt then F (x) = f (x). The chain rule tells us that, for a dierentiable function p(x), d dx Also,
a b p(x) a x
f (x) dx =
b
a
f (t) dt =
b
a
f (z) dz = . . ..
a a
If the interval is empty the integral is zero: Neighbouring integrals can be combined:
b a c c
f (x) dx = 0.
f (t) dt =
p(x) q(x)
d F (p(x)) = F (p(x))p (x) = f (p(x))p (x). dx
p(x) 0 q(x) 0
f (x) dx +
b
f (x) dx =
a
f (x) dx.
b a
f (t) dt =
f (t) dt
f (t) dt,
The order of the limits matters:
b
f (x) dx =
b
f (x) dx.
so that
The greater the function the greater the integral: If f (x) g(x) on [a, b] then
a
d dx
p(x) q(x)
f (t) dt = f (...