Young Won Lim6/25/15

Definite Integrals

Young Won Lim6/25/15

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Definite Integrals 3 Young Won Lim6/25/15

dy = f ' (x) dx

dydx

= f ' (x)

ratio not a ratio

dy =dfdx

dx

Differentials and Derivatives (1)

for small enough dx

differentials derivative

f (x1 + dx ) ≈ f (x1) + dy

= f (x1) + f ' (x1)dx

f (x1 + dx ) = f (x1) + dy

= f (x1) + f ' (x1)dx

f (x1 + dx ) − f (x1)

dx= f ' ( x1)

Definite Integrals 4 Young Won Lim6/25/15

dy = f ' (x) dx

dy =dfdx

dx

Differentials and Derivatives (2)

dy = f dx

dy = Dx f dx

∫ dy = ∫ dfdx

dx

∫ dy = ∫ f ' (x) dx

∫ dy =∫ 1dy = y

Definite Integrals 5 Young Won Lim6/25/15

Integration Constant C

∫ dy = ∫ dfdx

dx

∫ dy = ∫ f ' (x) dx

y + C1 = f (x) + C2

y = f (x) + C

place a constant

place another constant

∫ dy = ∫ dfdx

dx + C

differs by a constant

∫ dy = ∫ f ' (x) dx + C

y = f (x) + C

place only one constant from the beginning

Definite Integrals 6 Young Won Lim6/25/15

Applications of Differentials (1)

∫ f ( g(x ) )⋅g ' (x) dx ∫ f ( u ) du=

Substitution Rule

u = g( x) du = g ' (x )dx

∫ f (g)dgdx

dx ∫ f ( g ) dg=

du =dgdx

dx(I)

(II)

Definite Integrals 7 Young Won Lim6/25/15

Applications of Differentials (2)

∫ f (x )g ' (x) dx f (x )g(x ) − ∫ f ' (x )g(x ) dx=

Integration by parts

u = f (x)

v = g(x )

du = f ' (x ) dx

dv = g ' ( x)dx

∫ f (x )g ' (x) dx f (x )g(x ) − ∫ f ' (x )g(x ) dx=

∫udv u v − ∫ v du=

du =dfdx

dx

dv =dgdx

dx

Definite Integrals 8 Young Won Lim6/25/15

?

Anti-derivative

f (x)

differentiation derivative of ?

?Anti-differentiation

f (x)Anti-derivative of f(x)

Definite Integrals 9 Young Won Lim6/25/15

F ' (x) = f (x)

F (x)

Anti-derivative and Indefinite Integral

∫ f (x)dx

Anti-derivative without constantthe most simple anti-derivative

Indefinite Integral

F (x) + C the most general anti-derivative

∫ f (x)dx = F (x) + C

: a function of x

Definite Integrals 10 Young Won Lim6/25/15

Anti-derivative Examples

f (x)=x2

F1(x)=13

x3

F2(x)=13

x3 + 100

F3(x)=13

x3− 49

All are Anti-derivativeof f(x)

the most general anti-derivative of f(x)

13

x3 + C

≡ ∫ x2dxindefinite Integral of f(x)

differentiation

Anti-differentiation

Definite Integrals 11 Young Won Lim6/25/15

Indefinite Integrals

∫a

x1

1 dx

x1 − a

∫a

x1

d fd x

dx

f (x1) − f (a)

∫a

x

1 dx

x − a

∫ dx

x + C

∫−c

xd fd x

dx

f (x) − f (a)

∫ d fd x

dx

f (x) + C

∫ dy

y + C

given x1

a variable x indefinite integral

given x1

a variable x indefinite integral

Definite Integrals 12 Young Won Lim6/25/15

Indefinite Integrals via the Definite Integral

∫ f (x)dxindefinite integral of f(x)

∫a

x

f (t ) dtanti-derivative by the definite integral of f(x)

ddx∫a

x

f (t ) dt = f (x )

∫ f (x) dx = F (x) + C

∫a

x

f (t) dt = F (x) − F (a)

a common reference point : arbitrary

∫a

x

f (t ) dt

Definite Integrals 13 Young Won Lim6/25/15

Definite Integrals via the Definite Integral

∫x1

x2

f (t) dt

a common reference point : arbitrary

[ F (x) + c ]x1

x2 = F( x2)−F (x1) [ F (x) ]x1

x2 = F (x1) − F (x2)

Anti-derivative without constant

∫ f (x)dxindefinite integral of f(x)

∫a

x

f (t ) dtanti-derivative by the definite integral of f(x)

ddx∫a

x

f (t ) dt = f (x )

= ∫a

x1

f (t) dt + ∫a

x2

f (t ) dt

∫a

x

f (t ) dt

Definite Integrals 14 Young Won Lim6/25/15

Indefinite Integral Examples

f (x)=x2

=13

x3+ C∫ x2dxindefinite integral

of f(x)

∫0

x

f (x) dx = [ 13 x3]0

x

=13

x3

∫a

x

f (x) dx = [ 13 x3]a

x

=13

x3−

13a3

∫a

x

f (t ) dt = [ 13 t3]a

x

=13

x3−

13

a3

∫a

x

t 2 dt =13

x3−

13

a2anti-derivative by the definite integral of f(x)

ddx∫a

x

f (t ) dt = f (x ) = x2

Definite Integrals 15 Young Won Lim6/25/15

Definite Integrals on [a, x1]

∫a

x1

f ' (x) dx [ f (x) ]ax1 = f (x1)−f (a)

∫a

x1

g(x) dx [G (x)]ax1 = G (x1)−G(a)

view (I)

view (II)

∫a

x1

1 dx f ' (x) = 1

g(x) = 1

∫a

x1

f ' (x) dx

∫a

x1

1 dx ∫a

x1

g(x) dx

view (I)

view (II)

Definite Integrals 16 Young Won Lim6/25/15

Definite Integrals on [a, x1]

a x1

1

∫a

x1

1 dx

a x1

1

x1−a

∫a

x1

1 dx

dx

dy =dydx

dx = f ' (x)dx

f ' ( x)= 1 g(x) = 1

view (I) view (II)

=∫a

x1

f ' (x) dx

G( x1) = x 1 − a

=∫a

x1

g (x) dx

Definite Integrals 17 Young Won Lim6/25/15

Definite Integrals on [x1, x

2]

a x1

1

x2a x1 x2

x

area

length

f ' ( x)= 1 g(x) = 1

view (I) view (II) G( x) = x

[ f (x) ]x1

x2 = f (x2)−f (x1) [G(x)]x1

x 2 = G (x2)−G(x1)

arbitrary reference point (a, f(a))

arbitrary reference point (a, G(a))

Definite Integrals 18 Young Won Lim6/25/15

A reference point : integration constant C

x2 − x1

f (x) = x∫x1

x2

1 dx

= [ f (x)]x1

x2

= [ f (x) − f (a)]x1

x2

= [ f (x) + C ]x1

x2

arbitrary reference point (a, f(a))

x2 − x1

G (x) = x∫x1

x2

1 dx

= [G (x)]x1

x2

= [G (x) − G (a)]x1

x2

= [G(x) + C ]x1

x2

arbitrary reference point (a, G(a))

view (I) view (II)Anti-derivativewithout a constant

= ∫c

x2

f ' (x)dx −∫c

x1

f ' (x)dx = ∫c

x2

g(x )dx −∫c

x1

g(x)dx

f ' (x) g (x)

Anti-derivativewithout a constant

Definite Integrals 19 Young Won Lim6/25/15

Indefinite Integrals through Definite Integrals

∫ 1 dx

= f (x) − f (a) = x − a

∫a

x1

f ' (x) dx ∫ 1 dx

= G(x) − G (a) = x − a

∫a

x1

g(x) dx

a x1

x1−a

−a

a x1

x1−a

−a

G(x )= x − a

view (I) view (II)

= f (x) + C = G (x) + C

G(x )= x + C

x − a

x − a

f (x )= x − a

f (x )= x + C

arbitrary reference point (a, f(a))

arbitrary reference point (a, G(a))

Definite Integrals 20 Young Won Lim6/25/15

Definite Integrals on [x1, x

2]

a x2

f ' (x)

x1 a x2

G(x2)

x1

G(x1)

G(x )

area

length

∫x1

x2

f ' (x) dx ∫x1

x2

g (x) dxview (I) view (II)

Definite Integrals 21 Young Won Lim6/25/15

Definite Integrals on [a, x1] and [a, x

2]

a x2

a x2x1 a x2

G(x2)

x1

G(x1)

G(x )

length

a x2

G(x2)

x1

G(x1)

G(x )

length

f ' (x)

area

area

∫c

x2

f ' (x)dx −∫c

x1

f ' ( x)dx ∫c

x2

g(x )dx −∫c

x1

g(x)dx

Definite Integrals 22 Young Won Lim6/25/15

Indefinite Integrals through Definite Integrals

a x1a x1

G(x1)− G(a) y = G (x) − G(a)

f ' (x)

= f (x) − f (a) = x − a = G (x) − G(a) = x − a

= f (x) + C = G(x) + C

∫a

x1

f ' (x) dx ∫a

x1

g(x) dxview (I) view (II)

arbitrary reference point (a, f(a))

arbitrary reference point (a, G(a))

Definite Integrals 23 Young Won Lim6/25/15

Derivative Function and Indefinite Integrals

limh→0

f (x1 + h) − f ( x1)

hf ' (x1)

limh→0

f (x2 + h) − f (x2)

hf ' (x2)

limh→0

f (x3 + h) − f (x3)

hf ' (x3)

f ' (x ) = limh→0

f (x + h) − f (x)

h

x1 , x2 , x3

f ' (x1) , f ' (x2) , f ' (x3)

∫x1

x2

f (x ) dx

F (x ) + C = ∫a

x

f ( x) dx

[ x1 , x2] ,[ x3 , x4] , [x5 , x6]

[ F(x ) ]x 1

x 2 , [ F(x) ]x 3

x 4 , [ F (x ) ]x5

x6

∫x3

x4

f (x ) dx

∫x5

x6

f ( x ) dx

function of x function of x

Young Won Lim6/25/15

References

[1] http://en.wikipedia.org/[2] M.L. Boas, “Mathematical Methods in the Physical Sciences”[3] E. Kreyszig, “Advanced Engineering Mathematics”[4] D. G. Zill, W. S. Wright, “Advanced Engineering Mathematics”