short notes module 5 app integ

SHORT NOTES MODULE 5 : APPLICATIONS OF INTEGRATION

DEFINITE INTEGRAL

TERMINOLOGY

upper limit

integrand b

 f ( x ) dx a

lower limit

variable of integration

b

 f ( x ) dx = F (b) − F (a) a

Value of integration is a constant.

Definite integral of trigonometric/inverse trigonometric function must be evaluated in radian.

AREA AREA BETWEEN CURVES

dx

top function b

f(x) – g(x)

f(x)

Area, A =

 f ( x ) − g ( x ) dx a

bottom function g(x) a

b bottom function; g(x) = 0

dx

b

f(x) f(x)

a

Area, A =

 f ( x ) dx a

b

f(y)

right function

f(y) – g(y) d

d

Area, A =

dy g(y)

c

 f ( y ) − g ( y ) dy c

left function

f(y) d

left function; g(y) = 0

f(y) dy

c

d

Area, A =

 f ( y ) dy c

VOLUME OF SOLID REVOLUTION AREA ROTATES ABOUT X-AXIS

dx f(x) r

f(x) a

b b



Volume, V =   f ( x )  dx

DISK METHOD

disk volume = surface area  thickness

2

a

dx R

f(x) – g(x)

f(x)

r g(x) a

b

b

WASHER METHOD



Volume, V =   f ( x )  − g ( x )  dx a

washer volume = volume of bigger disk - volume of cut out disk

2

2

AREA ROTATES ABOUT Y-AXIS

f(y) f(y)

d

r dy

c

d

DISK METHOD



Volume, V =   f ( y )  dy 2

c

disk volume = surface area  thickness

f(y) f(y) – g(y)

R

d

r

dy c

g(y)

d

DISK METHOD



Volume, V =   f ( y )  − g ( y )  dy c

washer volume = volume of bigger disk - volume of cut out disk

2

2

MOMENT OF INERTIA REFERENCE AXIS : X-AXIS/PARALLEL TO X-AXIS

FIRST MOMENT OF AREA

first moment = distance ๏ ด area d

Qx =

๏ ฒ y dA = ๏ ฒ y ( xdy ) A

c

f(y) SECOND MOMENT OF AREA (MOMENT OF INERTIA)

x = f(y) d dy

second moment = distance ๏ ด area 2

d

๏ ฒ

(๐ ฅาง , ๐ ฆเดค)

๏ ฒ

I x = y2 dA = y 2 ( xdy ) A

c

c

MOMENT OF INERTIA ABOUT CENTROIDAL AXIS

Ix =

๏ ฒ (y โ y) A

2

d

dA =

๏ ฒ ( y โ y ) ( xdy ) 2

c

๐ ฆ = ๐ ฆเดค

REFERENCE AXIS : Y-AXIS/PARALLEL TO Y-AXIS

FIRST MOMENT OF AREA

first moment = distance ๏ ด area b

Qy =

๏ ฒ x dA = ๏ ฒ x ( ydx ) A

a

SECOND MOMENT OF AREA (MOMENT OF INERTIA)

second moment = distance2 ๏ ด area ๐ ฅ = ๐ ฅาง b

๏ ฒ

๏ ฒ

I y = x2 dA = x 2 ( ydx ) A

dx

a

f(x) MOMENT OF INERTIA ABOUT CENTROIDAL AXIS

Iy =

๏ ฒ(

xโ x

A

)

2

b

dA =

๏ ฒ (x โ x)

2

( ydx )

a

CENTROID

x=

Qy A

Q y= x A

( )

Centroid, C = x, y

a

y = f(x)

b