7/31/2019 Pure 17

1/4

x

yy = f(x)

a b

x

y

a b

y = g x

y

x = f(y)

a

b

y = g(x)

x

y

y = f(x)b

a

sr

X

r=f()

=1

O

=2

X

r=f()

=1

O

=2r=g()

97422441.doc Page 1 / 4

Pure Math Geom etric Applications of DeIntegrals

Areaof PlaneFiguresCartesian coordinate system:

The area under the equationy =f(x) in the regionx[a, b] is given by: =b

adxxfA )(

The area under thex-axis is negative whereas above the axis is positive The area bounded by the equationsy =f(x) andy =g(x) in the regionx[a, b] is given by:

=b

adxxgxfA )]()([

The area under the equationx =f(y) in the regiony[a, b] is given by: =b

adyyfA )(

The area at right-hand side of they-axis is positive whereas at left-hand side of the axis is

negative The area bounded by the equationsx =f(y) andx =g(y) in the regiony[a, b] is given by:

=b

adyygyfA )]()([

Parametric equations: When a curve is given by parametric equation:

==

)()(

tgytfx t

The area at the rangex[a, b] is:

====

=

2

1

)(')()(')(t

t

bx

ax

b

adttftgdttftgydxA

a =f(t1) i.e.x = a when t= t1 b =f(t2) i.e.x = b when t= t2

The absolute area at the rangex[a, b] is: ==2

1

)('|)(|||t

t

b

adttftgdxyA

Polar equations: Definite integrals can be used to find the area bounded by r=f()

and radius vectors =1, =2

For a sector with angle and radius rhas the area === 221

21

21 )( rrrrsA

The total area ofr=f() from =1 to =2 is: =

=

=

=

=2

1

2

1

)( 22

1

rAA

When taking the limit 0,

====

=

2

1

2

1

dfdrrA 2212

212

21

0)]([)(lim

2

1

The area enclosed by r=f(), r=g() and the radius vectors =1, =2:

( ) ==22

11

22

212

2

2

121 )]([)]([)( dgfdrrA

7/31/2019 Pure 17

2/4

y = f(x)

a

x

s

x

y

a b

x

x2x1

Page 2 / 4 97422441.doc

ArcLengthCartesian coordinate system: In the curvey=f(x), the length of the curve inx[a, b] iss In any partition of line ab, whose width is x and difference

inf(x) is

y, the portion of the curve is approximately equal

to: 22 )()( yxs +=

The total length of curve: =

=

=

=

+=bx

ax

bx

ax

yxss 22 )()(

By taking the limit x0,

+=

+=+=

=

=

=

=

b

a

bx

axx

bx

axx

dxdx

dy

x

yxyxs

22

0

22

011lim)()(lim

Length of curve: +=

+=

b

a

b

a

dxxfdxdx

dys 2

2

)]('[11

Parametric equations:

When a curve is given by parametric equation:

==

)()(tgytfx

t

The length of curve at the rangex[a, b] is:

( )

+=

+=

+=

1

2

1

2

2222

/

/11

t

t

t

t

b

a

dtdt

dy

dtdxdt

dtdx

dtdx

dtdydx

dx

dys

Length of curve: ( ) +=

+=

2

1

1

2

22

22

)]('[)]('[t

t

t

t

dttgtfdtdt

dy

dtdxs

a =f(t1) i.e.x = a when t= t1 b =f(t2) i.e.x = b when t= t2

Polar equations: The length of curve of the equation r=f() from =1 to =2:

Transformed to Cartesian coordinate:

====

sin)(sincos)(cos

fryfrx

Length of curve: ( ) +=

+=

2

1

1

2

22

22

)]('[

drrdd

dy

ddxs

Volumeof Solidsof RevolutionDisc method: Lety =f(x) be a cts function defined on [a, b] and the curve is

revolved about thex-axis At any partition [x1,x2] in [a, b] which has the width x, the volume

of revolution of the curve in this partition is like a disc Volume of a disc: V= r2h = y2(x) = [f(x1)]2x

Total volume of revolution: =

=

=

==

bx

ax

bx

axxxfVV 2)]([

When taking the limit x0,

====

=

b

a

b

a

bx

axx

dxxfdxyxxfV )()]([lim222

0

7/31/2019 Pure 17

3/4

y=f(x)

a b

h

y=f(

a

xy=g(x)

97422441.doc Page 3 / 4

The volume of revolution ofy =f(x) about the liney = h in theregionx[a, b] is:

==b

a

b

adxhxfdxhyV 22 ])([)(

The volume of revolution ofx =f(y) about they-axis in the regiony[a, b] is:

==b

a

b

adyyfdyxV 22 )]([

The volume of revolution ofx =f(y) about the linex = kin the regiony[a, b] is:

==b

a

b

adxkyfdykxV 22 ])([)(

Shell method: Lety =f(x) andy =g(x) be cts functions defined on [a, b]The area bounded byy =f(x),y =g(x),x = a,x = b is on the

xy-planeThe area is revolved about they-axis

At any partition in [a, b] of thex-axis, which has the width x, thestrip revolved is like a hollow cylinder Volume of the cylinder: V= (2x)[f(x) g(x)](x)

Total volume is: =

=

=

==

bx

ax

bx

ax

xxgxfxVV )]()([2

By taking the limit x0, ===

=

b

a

bx

axx

dxxgxfxxxgxfxV )]()([2)]()([lim20

Volume of revolution: ==b

a

b

adxyyxdxxgxfxV )(2)]()([2 21

y1 =f(x) y2 =g(x)y1 y2

Lety =f(x) andy =g(x) be cts functions defined on [a, b]The area bounded byy =f(x),y =g(x),x = a,x = b is on thexy-plane

Area bounded by curvesx =f(y) andx =g(y) in the regiony[a, b] is revolved about thex-axis

Volume of revolution: ==b

a

b

adyxxydyygyfyV )(2)]()([2 21

Solids w/known parallel cross section: For a solid which the cross-sectional area perpendicular to thex-axis is known The area at any positionx0 on thex-axis isA(x0)

The volume of the solid between positionsx = a andx = b is: =b

adxxAV )(

7/31/2019 Pure 17

4/4

Page 4 / 4 97422441.doc

Areaof Surfaceof RevolutionArea of surface of revolution: A curvey=f(x) in the regionx[a, b] is revolved about thex-axis In any partition in [a, b] on thex-axis, whose width is x, the curve is: 22 )()( yxs +=

When revolved, the area of the surface is A = 2ys =2

1)(2

+x

yxxf

Total area: =

=

+bx

ax

xx

yxfA

2

1)(2

By taking the limit x0

Total area:

+=

+=

=

=

b

a

bx

axx

dxdx

dyyx

x

yxfA

22

0121)(lim2

Area of the surface of revolution: +=+=b

a

b

a dxxfxfdxdxdyyA 2

2

)]('[1)(212