Application of Definite Integral

Applications of Definite Integrals Advanced Level Pure Mathematics

Prepared by Mr. K. F. Ngai

Page 1

Advanced Level Pure Mathematics

Applications of Definite Integrals

Areas 2

Arc Length 8

Volumes of Solids of Revolution 11

Area of Surface of Revolution 13

8



Page 2

Areas A Equations of Curves are represented in Rectangular Form

Let A denote the area ( or total area) of the shaded region.

Theorem The area enclosed by the graph of )x(fy = , the x-axis and the lines ax = and bx = is

equal to dx b

a)x(f or dx

b

ay .

Theorem The area enclosed by the graph of )y(gx = , the x-axis and the lines cy = and dy = is

equal to dy d

c)y(g or dy

d

cx .



Page 3

Example Find the area enclosed by the graph of 2xy = , the axis and the lines 1x = and 3x = . Solution

Example Find the area bounded by the following curves.

(a) Ellipse: 1b

y

a

x2

2

2

2

=+ .

(b) Cycloid:

=

=

)tcos1(ay

)tsint(ax, 2t0 and x-axis.



Page 4

Example Find the area enclosed by the graph 1xy2 += , the y-axis and the lines 2y = and 3y = .

Example Given a conic 09y8x6y4x:C 22 =+++ . (a) By completing squares and translation coordinate axes, transform the equation of

C to standard form. What is this curve? (b) find the area of the region bounded by C . ( Ans: 2 )

Solution



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B Equations of Curves are in parametric Form

It is known that the area between the curve )x(fy = and the lines ax = , bx = and 0y =

is given by dx b

ay .

If the equation of the curve is in parametric form

=

=

)t(Gy

)t(Fx,

where t is a parameter, and if ,bxwhent

;axwhent

==

==

)t('Fdt

dx= is a continuous function on ],[ , and )t('F does not change sign is in ),( ,

then the area of the region bounded by the curve

=

=

)t(Gy

)t(Fx, the x-axis and the lines

ax = , bx = is dt dt )t('xy)t('F)t(G

= . ( Integration by substitution )

This formula is also true when if > . In this case 0)t('Fdt

dx= for all ),(t .

Example Find the area of the ellipse

=

=

tsinby

tcosax, where 0a> , 0b > .

Example Consider the curve

=

=

t2y

2tx 2. Find the area bounded by the curve and y-axis.



Page 6

AL91II-3 Consider the curve

=

=

tcosy

tsinx3

3

,

2

t0 .

Find the area bounded by the curve, the x-axis and y-axis.

Solution The figure is symmetric about both of the coordinates axes.


=

=

tty

1tx3

2

. Find the area bounded by Loop of curve.



Page 7

Example Suppose that the curve in figure with parametric form )t(fx = , )t(gy = . If 1t and 2t are

the parameters of points A and B . Show the shaded area is dtdt

dx

dt

dy)yx(

2

1 2

1

t

t .



Page 8

Arc Lengths

A Equations of curves are in Rectangular Form

Theorem If a curve )x(fy = has a continuous derivative on ]b,a[ , then the length of the curve from

ax = to bx = is given by dxdx

dydx

+=+

b

a

2b

a

2 1))x('f(1 .

Remark If the equation of the curve is in the form )y(gx = , then the length of the arc between cy =

and dy = is given by dydy

dxdy

+=+

d

c

2d

c

2 1))x('g(1 .

Example Find the length of the parabolic arc x4y2 = from )0,0( to )4,4( .

Example Find the length of the loop of the curve 22 )x4(xy12 = .

Remark



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B Equations of Curves are in Parametric Form

Theorem When a function is expressed in parametric form )t(fx = and )t(gy = , the arc length s of the curve from at = to bt = is given by

dt +=b

a

22 ))t('g())t('f(s

Example Find the length of curve 2t4x = , 3t2y = between 1t,0t == .

Example Find the length of the loop of the curve 22 )xa3(xay9 = , 0a> .



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AL91II-3 Consider the curve

=

=

tcosy

tsinx3

3

,

2

t0 .

Find the length of the curve.

Example Find the total length of the curve:

=

=

,tsinay

tcosax3

3

where 0a> .

Question In the above example, as t increases from 0 to 2 , the point will move along the curve for a complete turn. Is it correct to find the arc length by evaluating the integral

2

0

22

0

22

2

tsina3

=

+

dtdt

dy

dt

dx ?

Always bear in mind that 022

+

dt

dy

dt

dx.



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Volumes of Solids of Revolution

A Disc Method

Theorem Let )x(fy = be a continuous function defined on ]b,a[ , and S be the region bounded by the curve )x(fy = , the lines ax = , bx = and the x-axis. Then the volume V of the solid generated by revolving the region S one complete revolution about the x-axis is given by

[ ] dxdx 2ba

b

a

2 )x(fyV == .

Remark In parametric form

=

=

)t(gy

)t(hx, the volume of solid of resolution generated by revolving the

region enclosed by the graph, x-axis and from 1t to 2t about x-axis.

dt)t('h))t(g(V2

1

t

t

2

=

Theorem Let )x(fy = be a continuous function defined on ]b,a[ , and S be the region bounded by the curve )x(fy = , the lines ax = , bx = and hy = . Then the volume V of the solid generated by revolving the region S one complete revolution about the hy = is given by

[ ] dxdx 2ba

b

a

2 h)x(f)hy(V ==

Example Find the volume of the solid generated by revolving one complete revolution of the upper half

region of the closed curve 1b

y

a

x 32

2

=

+

about the line by = , )0b,0a( >> .



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Example Find the volume of the solid generated by revolving the region bounded by the cycloid )tsint(ax = , )tcos1(ay = , where 0a> and 2t0 , and the x-axis about the

x-axis.

Example Find the volume of solid generated by revolving the region bounded by the curve tcosx 3= tsiny 3= about y-axis.

B Shell Method

Theorem Let f be a function continuous on ]b,a[ . If the area bounded by the graph of )x(f , the x-axis and the lines ax = and bx = is revolved about the y-axis, the volume V of solid

generated is b

a)x(xf2 dx

Example Derive a formula for evaluation the volume of a cylinder of a radius a and height h by using the shell method.



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Example Find the volume of the solid generated by revolving the region bounded by the curves

xxy 2 += , 2xey = and the lines ,1x = 2x = one complete revolution about the y-axis.

Area of Surface of Revolution

Theorem Suppose )x(fy = has a continuous derivative on ]b,a[ . Then the area S of the surface of revolution by the arc of the curve )x(fy = between ax = and bx = about the x-axis is

S = [ ] dx 2ba

)x('f1)x(f2 +

= dx

2b

a dx

dy1y2

+

Remark The corresponding formula for the area of the surface of revolution obtained by revolving an arc of a curve )y(gx = from cy = to dy = about the y-axis is

S = [ ] dy 2dc

)y('g1)y(g2 +

= dy

2d

c dy

dx1x2

+

Theorem If a portion of the curve of parametric equations )t(xx = , )t(yy = between the points corresponding to 1t and 2t is revolved about the x-axis, the surface area S is

S = dtds +=2

1

2

1

t

t

22t

t)]t('y[)]t('x[)t(y2)t(y2 .



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=

=

sinry

cosrx. Find the surface area of a sphere.

AL93II-4 Find the area of the surface obtained by rotating the following curve about the x-axis:

=

=

tcosy

tsinx3

3

,

2

t

2

.

Example Find the area of the surface generated by rotating the arc of the curve tsinex t= ,

,tcosey t= from 0t = to 2

t = about the x-axis.



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Example Find the area of surface generated by revolving the graph of 32 ty,t3x == between 0t = and 2t = about y-axis.

Example The curve C is represented by 2t4x = and tln4ty 4 = (a) Find the length of the loop of C from 1t = to 2t = .

(b) Find the Volume of the solid generated by revolving the arc in (a) about y-axis. Find also the surface area.

Application of Definite Integral

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