Change of Variables
- Prof. Caine
The definite integral on the left is equal to the definite integral on the right via the change of variables , or more abstractly, with and on [0,1]. Since both integrands are non-negative, we can interpret the integrals as measuring areas under the corresponding curves and the two areas do appear to be the same. Note that and so that the endpoints of the right hand integral are out of order. The resulting sign change to the integral in order to put the endpoints back in order is compensated by the fact that introducing a factor of minus 1 into the left hand integral as well. This applet attempts to illustrate the geometry of this equivalence using transformations of Riemann sums. For a given n, the LEFT(n) Riemann sum for over [0,1] is shown. The points of its partition are regularly spaced and are list from left to right. Under the change of variables , the image of this partition consists of irregularly spaced points which are then listed from right to left. Thus, the LEFT(n) Riemann sum in x for f(g(x))g'(x) is converted to a RIGHT(n) Riemann sum in u for f(u). The spacing between the points of the image partition is stretched by a factor depending on position of the sub-interval that it is associated to. Sub-intervals near zero in x correspond to shrunken sub-intervals in u near 1 and sub-intervals near 1 in x correspond to amplified sub-intervals near 0 in u. The negativity of the derivative of g accounts for the reversal in orientation from LEFT to RIGHT and the corresponding sign change of the integral needed to reorder the endpoints of integration. The derivative g'(x) accounts for the non-uniform change in scale of the bases of the rectangles. Thus, the heights must be determined by f(u).