Convert equation to cylindrical coordinates

This formula turns out to be a special case of a more general formula which can be used to evaluate multiple integrals. This is called the change of variable formula for integrals of single-variable functions, and it is what you were implicitly using when doing integration by substitution. In general, if x = g вЃў ( u ) is a one-to-one, differentiable function from an interval (which you can think of as being on the вЂњ u-axisвЂќ) onto an interval (on the x-axis), which means that g вЂІ вЃў ( u ) в‰ 0 on the interval ( c, d ), so that a = g вЃў ( c ) and b = g вЃў ( d ), then c = g - 1 вЃў ( a ) and d = g - 1 вЃў ( b ), and = в€« 0 3 1 2 вЃў ( u + 1 ) вЃў u вЃў рќ‘‘ u, which can be written as Then performing the substitution as we did earlier gives Then substituting that expression for x into the function f вЃў ( x ) = x 3 вЃў x 2 - 1 givesДЇ вЃў ( x ) = f вЃў ( g вЃў ( u ) ) = ( u + 1 ) 3 / 2 вЃў u , That is, on we can define x as a function of u, namely On the interval of integration, the function x в†¦ x 2 - 1 is strictly increasing (and maps onto ) and hence has an inverse function (defined on the interval ). Let us take a different look at what happened when we did that substitution, which will give some motivation for how substitution works in multiple integrals. Recall that if you are given, for example, the definite integral

Convert equation to cylindrical coordinates