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Change of variables

Change of Variables Change of variables is a technique used in multivariable calculus to transform an integral into an equivalent integral with a simpler var...

Change of Variables#

Change of variables is a technique used in multivariable calculus to transform an integral into an equivalent integral with a simpler variable. This allows us to solve integrals that would otherwise be impossible or very difficult to solve.

Key idea: We change the variable of integration to make it simpler and express it in terms of the original variable.

Steps of change of variables:

Identify: We first identify the variable(s) to change and the variable(s) to which they will be changed.
Substitute: We substitute the old variable(s) into the new variable(s) in the integrand.
Simplify: We simplify the integrand by performing necessary manipulations, such as combining like terms or factoring expressions.
Change the limits: We need to adjust the limits of the integral based on the variable transformation.

Examples:

Example 1:

Original integral:

$\int_0^2\int_0^1x^2ydxdy$

Substitution:

$u = x^2$

$dv = ydx$

Substituting, we get:

$\int_0^4\int_0^1u(v)^{1/2}dvdu$

Simplifying and evaluating:

$\int_0^4\left[\frac{2}{3}u^{3/2}\right]_0^1du = \frac{2}{3}$

Example 2:

Original integral:

$\int_1^3\int_0^x\frac{1}{x}dxdy$

Substitution:

$v = x$

Substituting, we get:

$\int_1^3\left[\frac{1}{v}y\right]_0^xdy = \int_1^3xdx = 3$

Key points:

Change of variables is a versatile technique applicable to various types of integrals.
It allows us to simplify complex integrals and solve them more easily.
The key is to identify the variables to change, substitute them, simplify the integrand, and adjust the limits of the integral.

Remember: Practice and patience are crucial for mastering change of variables