TOPIC

LENGTH OF AN ARCand

AREA OF SURFACE OF REVOLUTION

LENGTHS OF CURVES

To find the length of the arc of the curve y = f (x) between x = a and x = b let ds be the length of a small element of arc so that:

2

222 1 thus

dx

dy

dx

dsdydxds

LENGTHS OF CURVES

In the limit as the arc length ds approaches zero:

and so: 2

1ds dy

dx dx

2

1

b

x a

b

x a

dss dx

dx

dydx

dx

LENGTHS OF CURVES – PARAMETRIC EQUATIONS

Instead of changing the variable of the integral as before when the curve is defined in terms of parametric equations, a special form of the result can be established which saves a deal of working when it is used. Let:

dt dtdy

dtdx

s and dtdy

dtdx

dtds

0dt as so dtdy

dtdx

dtds

so

dydxds before As .tFx and tfy

t

tt 1

2 2222

222

222

LENGTH OF AN ARC

A. Rectangular Coordinates

f(x)y , 12

dxdx

dyds g(y) x, dy

dy

dx1ds

2

B. Parametric Form

dtdt

dy

dt

dxds

22

when x=x(t), y=y(t); where t is a parameter

1. 2.

C. Polar Coordinates

f(r) , drdr

dr1ds

22

)g(r , dd

drrds

22

2.1.

EXAMPLE

Find the length of the arc of each of the following:

2

3

3

3

ty

ttx

1.

from t=o to t=1

2.

tsiney

tcosext

t

from t=o to t=4

5. Length of the arc of the semicircle222 ayx

2x to 1x fromee

lny x

x

1

13.

4.),(to),(from

xy2100

8 23

AREA OF SURFACE OF REVOLUTION

• DEFINITION:Let y = f(x) have a continuous derivative on the interval [a, b]. The area S of the surface of revolution formed by revolving the graph of f about a horizontal or vertical axis iswhere r(x) is the distance between the graph of f and the axis of revolution.

xof function a is y dx)x('f)x(rSb

a 212

If x = g(y) on the interval [c, d], then the surface area is

where r(y) is the distance between the graph of g and the axis of revolution.

yof function a is x dy)y('g)y(rSd

c 212

EXAMPLE

1. Find the area formed by revolving the graph of f(x) = x3 on the interval [0,1] about the x-axis.

2. Find the area formed by revolving the graph of f(x) = x2 on the interval [0, ] about the y – axis.

3. Find the area of the surface generated by revolving the curve

, 1 ≤ x ≤ 2 about the x – axis.4. The line segment x = 1 – y, 0 ≤ y ≤ 1, is revolved about the y

– axis to generate the cone. Find its lateral surface area.

2

xy 2