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MAT1322
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Termeh Kousha
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Lecture

Description

Integration and Its applications
Professor: Termeh Kousha
MAT 1322-Winter 2013 CONTENTS 2
Contents
1 Review of integration 3
1.1 Common Inde▯nite Integrals . . . . . . . . . . . . 3
1.1.1 Rules for Integrals . . . . . . . . . . . . 4
1.1.2 Integration by Substitution . . . . . . . . 5
1.1.3 Integration by Parts . . . . . . . . . . . .6
1.2 Finding the Area Between Two Curves . . . . . . 9
1.3 The Fundamental Theorem of Calculus . . . . . . 10
1.4 Area Between Intersecting Curves . . . . . . . . . 13
2 Improper Integrals 15
2.1 Improper Integrals of type I and II . . . . . . . . 15
2.2 The comparison Test . . . . . . . . . . . . . . . . 22 1 REVIEW OF INTEGRATION 3
1 Review of integration
0
Suppose that F(x) and f(x) are functions such that F (x) =
f(x) then we say that F(x) is an antiderivative of f(x).
For any antiderivative, F(x), of a function f(x), we use the
notation Z
f(x) dx = F(x) + C
R
where we say that f(x) dx is the inde▯nite integral (or gen-
eral antiderivative) of f(x) with respect to x and we call C the
constant of integration.
1.1 Common Inde▯nite Integrals
1. For any constant k,
Z
k dx = kx + C:
2. For any real number n except -1,
Z
n 1 n+1
x dx = x + C:
n + 1
3. Z
1 dx = lnjxj + C:
x
4. Z
e dx = e + C: 1 REVIEW OF INTEGRATION 4
1.1.1 Rules for Integrals
1. For any constant k and function f(x),
Z Z
kf(x) dx = k f(x) dx:
2. For any functions f(x) and g(x),
Z Z Z
(f(x) + g(x)) dx = f(x) dx + g(x) dx:
R
Ex: Find each of the following inde▯nite integral 5e ▯3x +▯2
▯1
7x dx.
0 3x
Ex: Given that f (x) = e + 4 and f(0) = 2, ▯nd the function
f(x). 1 REVIEW OF INTEGRATION 5
1.1.2 Integration by Substitution
Integration by Substitution for Inde▯nite Integrals:
1. Choose u = g(x) such that you can see g (x) in the inte-
grand.
2. Compute the di▯erential du = g (x) dx.
3. Substitute u and du into the integrand { there must not be
any x or dx remaining.
4. Solve the resulting simpler integral.
5. Back-substitute u = g(x) into the ▯nal result.
R
We use integration by substitution for f(x) dx when f(x) is
contains a function (g(x)) and its derivative (as a factor).
Ex: Z
5 3x + 6
2 dx
2 x + 4x + 1 1 REVIEW OF INTEGRATION 6
1.1.3 Integration by Parts
Integration by Parts Formula:
For inde▯nite integrals:
Z Z
u dv dx = uv ▯ du v dx
dx dx
For de▯nite integrals:
Z b dv Z bdu
u dx = [uv]a▯ v dx
a dx a dx
Rules of Thumb:
1. Choose u to be a function that gets no more complicated
when you take its derivative.
dv
2. Choose dx to be a function that gets no more complicated
when you take its integral.
Ex: Find
Z
xe dx: 1 REVIEW OF INTEGRATION 7
Notes:
▯ It is crucial to make the right choice. Making the wrong
0
choice for u and v will usually give you a harder integral
than the one you started with. For example, say you were
asked to solve Z
xe dx
and chose
u = ex v = x
giving you
x2
u = e x v =
2
Then integration by parts would give you
Z 2 Z 2
xe = x e ▯ x e dx
2 2
| {z }
even harder
We ended up with a worse integral, so we know we made
the wrong choice for u and v .
▯ Sometimes you have to do integration by parts twice. For
example, say you were asked to solve
Z
x e dx
The reasonable thing to do would be to choose
2 0 x
u = x v = e
giving you
u = 2x v = ex 1 REVIEW OF INTEGRATION 8
Then integration by parts gives you
Z Z
x e dx = x e ▯ 2xe dx
The second integral we can now do, but it also requires
parts. We take
u = 2x v = e x
giving us
0 x
u = 2 v = e
So we have
Z ▯ Z ▯
2 x 2 x x x 2 x x x
x e dx = x e ▯ 2xe ▯ 2e dx = x e ▯2xe +2e +C
Rn general, you need to do n integration by parts to evaluate
x e dx.
▯ In the case of de▯nite integrals, the integration by parts
formula becomes
Z Z
b ▯ b
uv dx = uv ▯ ▯ u vdx
a a a
Ex: Find Z
ln(x) dx: 1 REVIEW OF INTEGRATION 9
1.2 Finding the Area Between Two Curves
Suppose that you have two curves, y = f(x) and y = g(x) and
some closed interval [a;b] such that
▯ f(x) and g(x) are continuous on [a;b] and
▯ f(x) ▯ g(x) on [a;b].
Picture:
Question: How can we calculate the area between these two
curves on the interval [a;b]?
The area enclosed between two curves f(x) and g(x), where
f(x) ▯ g(x) is given by
Z b
A = (f(x) ▯ g(x)) dx:
a 1 REVIEW OF INTEGRATION 10
1.3 The Fundamental Theorem of Calculus
For a function, f(x), which is continuous on the interval [a;b],
we call Z b
f(x) dx
a
the de▯nite integral of f(x) with respect to x from a to b. The
numbers a and b are called the lower limit of integration and
upper limit of integration respectively.
Notice that, if f(x) ▯ 0 for all x in [a;b] then
Z b Z b
f(x) dx = f(x) ▯ 0 dx
a a
Rb
so a f(x) dx is simply the area between the curve y = f(x) and
the line y = 0 (the x-axis) from the vertical line x = a to the
vertical line x = b.
The Fundamental Theorem of Calculus: If f(x) is a contin-
uous function on the interval [a;b] and F(x) is any antiderivative
of f(x) t

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