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Change of variables

The limits of integration are often not easily interchangeable (without normality or with complex formulae to integrate). One makes a change of variables to rewrite the integral in a more "comfortable" region, which can be described in simpler formulae. To do so, the function must be adapted to the new coordinates.

Example 1a. The function is f (x, y) = (x − 1) + √ y; if one adopts the substitution u = x − 1, v = y therefore x = u + 1, y = v one obtains the new function f (u, v) = (u) + √ v.

There exist three main "kinds" of changes of variable (one in R, two in R); however, more general substitutions can be made using the same principle.