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