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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

❶

fxydy

ygx

ygx(,)

()

()

11

22

=

=

∫

[at constant x → u(x)]

❶

fxyzdx

xgyz

xgyz (,,)

(,)

(,)

11

22

=

=

∫

[at constant y, z → u(y, z)]

❷ uxdx Ub Ua

a

b() ()

()

=−

∫

U = anti-derivative of u

❷

uyzdy

yhx

yhx(,)

()

()

11

22

=

=

∫

[at constant z → v(z)]

❸ vzdz Vb Va

a

b() ()

()=−

∫

Note: The process which is

the inverse of differentiation is

termed anti-differentiation or

integration • Every continuous

function f(x) has an

anti-derivative which is the

integral of f(x)

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