Integration with change of variables
If is a diffeomorphism, the following theorem holds
The most common use of the above method are polar coordinates which are defined by the diffeomorphism . Thus, .
E.g. let us calculate
If we want to calculate the volume of a set we can do it using one of the following three methods:
where is the basis is the height at and is the area of the section for given .
Assume that we would like to calculate the volume of a pyramid with a base in form of a isosceles right triangle (with boundary of axes and line ) and height , so in point the height is given by the following formula . Therefore the section at has area , so the volume is:
which is consistent with our knowledge about geometry.
Length of a curve
We can also calculate a length of a curve using integrals. Given a parametrization of a curve , its length between and , is given by the following formula:
Let us check this formula for the line between and . Such a line has parametrization . Therefore its length is: