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Reference Guide

# Permachart - Marketing Reference Guide: Iterated Integral, Centroid, Multiple Integral

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1INTEGRAL CALCULUS A-834-6
AVERAGE VALUE THEOREM
If f(x) is continuous over [a,b], then
fx ba fxdx y
a
b
() ()==
1
which means that the integral
equals the area of the rectangle
()•()( )•ba fx ba y−=
FUNDAMENTAL THEOREMS
If f is continuous on [a, b], then
Fx ftdt
a
x
() ()=
means d
dx ftdt fx
a
x()
()
=
(that is, F is the anti-derivative of f)
If F is the anti-derivative of f, then
fxdx Fx Fb Fa
a
b
a
b
() () ()
()
=
[]
=−
If f’ is continuous, then
ftdt fx fa'
a
x() ()
()
=−
(that is, if you first differentiate f
and then integrate from a to x,
the result will differ from f(x) by at
most f(a))
If a is chosen so that f(a) = 0, then
differentiation and integration
exactly cancel each other out
AVERAGE VALUE OF A
FUNCTION
If f(x) = continuous over [a, b], then
yydx
dx
x
a
b
a
b
=
THEOREMS
TM
permacharts
Integral Calculus
INDEFINITE INTEGRALS
If F(x) + C has the derivative f(x), then F(x) + C is the anti-derivative or the indefinite integral of f(x)
= anti-derivative or indefinite integral
where: C = integration constant f(x) = integrand F(x) = particular integral
Example:
DEFINITE INTEGRALS
DOUBLE INTEGRALS
The limit of the sum of inscribed or circumscribed prisms with base DA = DxDy
equals the volume under the surface z = f (x, y) and is given by the double
integral between (x1, x2) and (y1, y2)
TRIPLE INTEGRALS
Triple integrals are three-dimensional extensions of double integrals defined in analogous fashion
ITERATED & PARTIAL INTEGRALS
For evaluation purposes, double and
triple integrals are expressed in iterated
forms of interchangeable order
• Each iteration is called a partial integral
TYPES OF INTEGRALS
f(x)
ab
x
y = f(x)
yyds
ds
s
a
b
a
b
=
= Average ordinate
with respect to
abscissa x over [a,b]
= Average ordinate
with respect to arc
length s over [a,b]
Givenderivative,then
d
dx Fx Cf
xF
x() () ()+
[]
== +CCfxdx
[]
=()
Givenderivative, then
d
dx xC
xx
C
43 4
4+
== +
=43
xdx= anti-derivative or indefinite integral
Definition 1
(a,b = lower and upper limits of integration x = variable of integration)
Definition 2 The limit of the sum of inscribed or circumscribed rectangles
equals the area under f(x) between a and b and is given by the
definite integral between a and b
Giventhe indefinite integral ()fxdx Fx
=()++
=+
[]
−+
[]
=−
C
fxdx Fb CFaCFb F
a
b
then () () () () (aa)
wherefxd
xa
a
b()
=definite integral from to bbfxdx
ab
of ()
lowerand upper limitsofi,=nntegration variable of integrationx =
lim () ()
ni
n
ia
b
fx xfxdx
→∞
=
1
f(x)
ab
x
xxx
f(x)
z
x
x1
x2
y1
y2
y
A
z = f(x,y)
lim (,,) (,,)
niii
n
ix
x
y
y
fx yz Vfxyzd
→∞ ∫∫
=
11
2
1
2zzdydx
z
z
1
2
lim (,
)(
,)
nii
n
ix
x
y
y
fx yA fxydydx
→∞ ∫∫
=
11
2
1
2
x
x
y
y
y
y
fxydydxfxydy dx
1
2
1
2
1
2
∫∫
=
(,)(,)
xx
x
x
x
y
y
y
fxydxdy
fxydy
1
2
1
2
1
2
∫∫
=
(,)
(,)
11
2
1
2
y
x
xfxydx
∫∫
=Partial,(,)IIntegrals
Double Integral Iterated Integral Iterated Integral
Double Integral Triple Integral
fxydydx
gx
gx
a
b(,)
()
()
1
2
fxyzdydxdz
gyz
gyz
hz
hz
a
b(,,)
(,)
(,)
()
()
1
2
1
2
Evaluate in 2 Steps Evaluate in 3 Steps
fxydy
ygx
ygx(,)
()
()
11
22
=
=
[at constant xu(x)]
fxyzdx
xgyz
xgyz (,,)
(,)
(,)
11
22
=
=
[at constant y, zu(y, z)]
uxdx Ub Ua
a
b() ()
=−
U = anti-derivative of u
uyzdy
yhx
yhx(,)
()
()
11
22
=
=
[at constant zv(z)]
vzdz Vb Va
a
b() ()
()=−
Note: The process which is
the inverse of differentiation is
termed anti-differentiation or
integrationEvery continuous
function f(x) has an
anti-derivative which is the
integral of f(x)
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