Calculus PresentationS.V.I.T.

Mechanical 2Semester 1

Topics :-i) Change of orderii) Change of variables

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Applicable when one integral is either difficult or impossible to evaluate, whereas the other Integral can be evaluated easily.

The change from one integral to the other is called change of order of integration.

For changing the order, we should sketch the region of integration. From the sketch, the new limits of integration can be determined

as usual.

i) Change of order of Integration

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( , )

( , )

x xx y x y x yu v

y yu v u v v u

u v

ii) Change of Variables using Jacobian

The Jacobian of the transformation T given by x = g(u, v) and y = h(u, v) is:

( , )

( , )

x yA u v

u v

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Replace x , y by their equivalent relations in terms of u & v, the area element (dx dy) by (J du dv) and the region R of integration in xy-plane by region in the uv-plane.

Working Rule :-

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Use the change of variables x = u2 – v2, y = 2uv to evaluate the integral where R is the region bounded by x-axis, parabolas y2 = 4 – 4x and y2 = 4 + 4x, y ≥ 0.

Example:-

R

y dA

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2 2

2 2( , )

2 2( , )

4 4 0

x xu vx y u v

y y v uu v

u v

u v

1 1 2 2

0 0

1 1 3 3

0 0

1 14 2 31 14 2 00

1 13 2 4

00

( , )2

( , )

(2 )4( )

8 ( )

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(2 4 ) 2

R S

u

u

x yy dA uv dA

u v

uv u v du dv

u v uv du dv

u v u v dv

v v dv v v

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By:-Group 3

1. 13BEMEG1042. 13BEMEG1053. 13BEMEG1064. 13BEMEG1075. 13BEMEG108

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