further differentiation and integration

8/10/2019 FURTHER DIFFERENTIATION AND INTEGRATION

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FURTHER DIFFERENTIATION

AND INTEGRATION


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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi

DIFFERENTIATION AND INTEGRATION OFHYPERBOLIC FUNCTION

The basic hyperbolic function in terms of and is xe xe

sinh , cosh2 2

x x x x

e e e e x x

+= =

Derivative formulas for hyperbolic function can be obtainedby first expressing the function in terms above.


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Example: Diff and int of hyperbolic
http://hyperbolic%20function.doc/http://hyperbolic%20function.doc/


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DIFFERENTIATION AND INTEGRATION OFINVERSE TRIGONOMETRIC FUNCTION ANDINVERSE HYPERBOLIC FUNCTION

Inverse trigonometric function

The notation are reserved exclusively forthe inverse trigonometric functions .Please take note that

1 1sin , tan , x x

1 1sin (sin ) x x


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Do You Remember!

Example: Diff and int of inv trigo
http://1_inverse%20trigo.doc/http://1_inverse%20trigo.doc/http://inverse%20trigo.doc/http://inverse%20trigo.doc/http://1_inverse%20trigo.doc/http://1_inverse%20trigo.doc/http://1_inverse%20trigo.doc/


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Inverse hyperbolic function

List of differentiation and integration of inverse hyperbolicfunction:

Example: Diff and int inv hypDo You Remember!
http://inverse%20hyperbolic.doc/http://2_inverse%20hyperbolic.doc/http://2_inverse%20hyperbolic.doc/http://2_inverse%20hyperbolic.doc/http://inverse%20hyperbolic.doc/


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Hyperbolic substitution is another technique for evaluatingintegral containing radical by making substitution involvinghyperbolic functions.Consider the table below

INTEGRATION BY HYPERBOLIC SUBSTITUTION

Example: Int by hyperbolic substitution
http://hyperbolic%20subs.doc/http://hyperbolic%20subs.doc/http://hyperbolic%20subs.doc/


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APPLICATION OF INTEGRATION

Arc length Surface area of

revolution

Curvature

------------------------------------------------------------------------- Arc length

If y=

f( x

)is a

smooth curve on,then the

arc length is

2

1

2

1 x

x

dy L dxdx = +

1 2[ , ] x x x=

If x = g ( y) is asmooth curve on

,then thearc length is

1 2[ , ] y y y=

2

1

2

1 y

y

dx L dydy = +

Parametric curveLet x = x (t ) and y = y(t ), and

, the arclength is

1 2[ , ]t t t =

2

1

2 2t

t

dx dy L dt

dt dt

= +

Example: Arc length
http://arc%20length.doc/http://arc%20length.doc/


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DEPARTMENT OF MATHEMATIC S AND STATISTICS, FSTPi

Surface area of revolution

If f ( x) is a smooth, nonnegative function on ,then the surface area of revolution that is generated byrevolving the portion of f ( x) on about the x-axisis

1 2[ , ] x x x=

1 2[ , ] x x x=

( )2

1

2

2 ( ) 1 ( )

x

xS f x f x dx = +If g ( y) is a smooth, nonnegative function on ,then the surface area of revolution that is generated byrevolving the portion of g ( y) on about the y-axisis

1 2[ , ] y y y=

1 2[ , ] y y y=

( )2

1

22 ( ) 1 ( )

y

y

S g y g y dy = +

Example: Surface area of revolution
http://surface.doc/http://surface.doc/http://surface.doc/


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Curvature

Curvature is a problem of determining how sharply a curvebends.Curvature of y = f ( x) denoted by (kappa - Greek) isdefined by

2

2

3/22

1

d ydx

dydx

= +


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Parametric curveLet x = x(t ) and y = y(t ), then curvature of parametric curvegiven by

where

3/22 2

| | xy yx

x y

= +

2 2

2 2, , ,dx dy d x d y

x y x ydt dt dt dt

= = = =

Example: Curvature

Radius of curvature

Radius of curvature, 1

=
http://curvature.doc/http://curvature.doc/

further differentiation and integration

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