Mathematics 1

geometry that for y= x 3/2

it should be slightly longer

Using the formula: L= we have:

L= = =

Making substitution u= we have: du= dx; x=0,u=1; x=1;u=;Then L= = u= 1.4397

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Curve Length - Example 1.59.

The cable which is strung between the two supports of the main span

on the Forth Road Bridge is approximately in the shape of a Catenary Curve. This is the shape that a chain hangs in between two supports. The supports are 1006m apart, and the cable is at height 0 in the middle and height 90m at the supports.

• The height of the cable is then
• h(x) = a cosh(x/a) − a, a = 1400m
• where x = 0 is the middle of the span, and x = ±503m are the two ends. What is the length of the cable?

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Forth Road Bridge, Scotland

Curve Length - Example 1.59- solution

• The given equation of the curve is:
• h(x) = a cosh(x/a) − a, a = 1400m
• The length of the curve : L=
• h'(x)=sinh(x/a)= sinh(x/a), then: L= =

= =

• (cosh2(u)-sinh2(u)=1, hence=cosh2())
• = 1400sinh(x/1400) = 1400(sinh(503/1400)+ sinh(503/1400))=2800 sinh(503/1400) = 1027.783m

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Curve Length – Example 1.60

• Find the length of the curve defined by f(x) = 3 − 3x2 between x = −1 and x = 1.
• Solution:
• f'(x) =− 6x

L= = ….(1)

Take at first the indefinite integral

( here a=6) Use substitution:

ax=sinh(u)⇒ dx=cosh(u)du; Using the formula: cosh2u-sinh2u=1 we get: =

Using double angle formula we get:

= =(u+sinh2u)+C …………….(2)

For calculating the definite integral (1) we should change its limits: x=-1 u=sinh-1(6x) .Hence

== (u+sinh2u) = =(2.492+sinh(4.984))=6.498

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Curve Length – Example 1.60 – obtaining the formula from the textbook

In the previous slide we’ve got the formula (2) :

= =(u+sinh2u)+C

Here u = sinh-1(ax)

For the second term we use the formula: sinh2 2sinhcoshsinh

Plugging in sinh(u) =ax we get:

sinh2 =2ax

And finally we get the formula from your textbook:

=(sinh-1(ax))+ +C………..(3)

Using (3) we can calculate the original integral (1) directly:

= (sinh-1(6x)+6x) = =(2.492+6)=6.498 - of course, it is the same result as we’ve got in the previous slide and computer calculus package (like Maple) gives also the same result

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