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Change of variable in double integral converted

Change of variable in double integral converted

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Change of Variables in Double Integral • Sometimes the evaluation of a double integral or triple integral with the given form may not be convenient to evaluate. • By choice of a suitable co-ordinate system (polar or spherical ), a given integral can be transformed into a simple integral involving the new variables. Transformation of co-ordinates x = f (u , v) and Let y = g (u , v) be the relations between the old variables (x,y) with the new variables (u,v) of the new co-ordinate system Then the given double integral will reduced to  F ( x, y) dxdy =  F ( f , g ) R R x u where J = y u x v y v J dudv is called the Jacobian of co-ordinate system Change of variables from Cartesian co-ordinates to Polar co-ordinates In this case u = r and v =  x = r cos  and y = r sin  x  ( x, y ) r J= =  (r ,  ) y r x  y  = cos  − r sin  sin  r cos  = r cos 2  + r sin 2  = r (sin 2  + cos 2  ) =r J = r cont… Hence  F ( x, y) dxdy =  F (r cos  , r sin  ) J drd R R =  F (r cos  , r sin  ) rdrd R Similarly from polar to Cartesian coordinates can described Solution cont… cont… cont… Solution cont… cont… Solution cont… cont… cont… cont… Solution cont… cont… cont… cont… cont… cont… Name: Description: ...
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