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3.
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The
challenging part in changing variables from (x,y) to (u,v) is to find the
transformation x = x(u,v) and
y = y(u,v) then using this information to
calculate the Jacobian, JT(u,v).
Use
the Jacobian to rewrite the transformed integral as follows:
I
= ∫ ∫
F(x,y) dA = ∫
∫ JT(u,v) du
dv
R S
One procedure is as follow:
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Step
1 |
Graph
the region, R, defined by x - f1(x)
= C1, x - f1(x)
= C2
and by x - f2(x) = D1, x
- f2(x) = D2.
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Step
2 |
Determine
the coordinates of the vertices for each corner of the region
in
R. |
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Step
3 |
Find
the new variables, u and v, by setting u = f1(x) and
v = f2(x). |
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Step
4 |
Determine
the coordinates of the vertices for each corner of the region
in S by substituting
the values determined in step 2. |
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Step
5 |
Plot
the new region of integration, S, in the u-v plane. Two objectives
are
to simplify the region of integration and to determine the limits of
integration in the u-v plane. |
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Step
6 |
Calculate
the Jacobian, JT(u,v) by taking the partial derivatives
∂x/∂u,
∂x/∂v, ∂y/∂u, and ∂y/∂v . |
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Step
7 |
Use
the Jacobian to evaluate the integral, I, as given above. |
Click
here for an example.
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