For the integral 60° $* f(x,y)dxdy the change of order is
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Answer:
Given a double integral
∬Df(x,y)dA
∬Df(x,y)dA
of a function f(x,y)f(x,y) over a region DD, you may be able to write it as two different iterated integrals. You can integrate with respect to xx first, or you can integrate with respect to yy first. If you integrate with respect to xx first, you will obtain an integral that looks something like
∬Df(x,y)dA=∫□□(∫□□f(x,y)dx)dy,
∬Df(x,y)dA=∫◻◻(∫◻◻f(x,y)dx)dy,
and if you integrate with respect to yy first, you will obtain an integral that looks something like
∬Df(x,y)dA=∫□□(∫□□f(x,y)dy)dx.
∬Df(x,y)dA=∫◻◻(∫◻◻f(x,y)dy)dx.
We often say that the first integral is in dxdydxdy order and the second integral is in dydxdydx order.
One difficult part of computing double integrals is determining the limits of integration, i.e., determining what to put in place of the boxes □◻ in the above integrals. In some situations, we know the limits of integration the dxdydxdy order and need to determine the limits of integration for the equivalent integral in dydxdydx order (or vice versa). The process of switching between dxdydxdy order and dydxdydx order in double integrals is called changing the order of integration (or reversing the order of integration).
Changing the order of integration is slightly tricky because its hard to write down a specific algorithm for the procedure. The easiest way to accomplish the task is through drawing a picture of the region DD. From the picture, you can determine the corners and edges of the region DD, which is what you need to write down the limits of integration.
We demonstrate this process with examples. The simplest region (other than a rectangle) for reversing the integration order is a triangle. You can see how to change the order of integration for a triangle by comparing example 2 with example 2' on the page of double integral examples. In this page, we give some further examples changing the integration order.
Example 1
Change the order of integration in the following integral
∫10∫ey1f(x,y)dxdy.
∫01∫1eyf(x,y)dxdy.
(Since the focus of this example is the limits of integration, we won't specify the function f(x,y)f(x,y). The procedure doesn't depend on the identity of ff.)
Solution: In the original integral, the integration order is dxdydxdy. This integration order corresponds to integrating first with respect to xx (i.e., summing along rows in the picture below), and afterwards integrating with respect to yy (i.e., summing up the values for each row). Our task is to change the integration to be dydxdydx, which means integrating first with respect to yy.
We begin by transforming the limits of integration into the domain DD. The limits of the outer dydy integral mean that 0≤y≤1,0≤y≤1, and the limits on the inner dxdx integral mean that for each value of yy the range of xx is 1≤x≤ey.1≤x≤ey. The region DD is shown in the following figure.
Change order of integration example region with exponential, x first
The maximum range of yy over the region is from 0 to 1, as indicated by the gray bar to the left of the figure. The horizontal hashing within the figure indicates the range of xx for each value of yy, beginning at the left edge x=1x=1 (blue line) and ending at the right curve edge x=eyx=ey (red curve).
We have also labeled all the corners of the region. The upper-right corner is the intersection of the line y=1y=1 with the curve x=eyx=ey. Therefore, the value of xx at this corner must be e=e1=ee=e1=e, and the point is (e,1)(e,1).
To change order of integration, we need to write an integral with order dydxdydx. This means that xx is the variable of the outer integral. Its limits must be constant and correspond to the total range of xx over the region DD. The total range of xx is 1≤x≤e1≤x≤e, as indicated by the gray bar below the region in the following figure.
Change order of integration example region with exponential, y first
Since yy will be the variable for the inner integration, we need to integrate with respect to yy first. The vertical hashing indicates how, for each value of xx, we will integrate from the lower boundary (red curve) to the upper boundary (purple line). These two boundaries determine the range of yy. Since we can rewrite the equation x=eyx=ey for the red curve as y=logxy=logx, the range of yy is logx≤y≤1logx≤y≤1. (The function logxlogx indicates the natural logarithm, which sometimes we write as lnxlnx.)
In summary, the region DD can be described not only by
0≤y≤11≤x≤ey
0≤y≤11≤x≤ey
as it was for the original dxdydxdy integral, but also by
1≤x≤elogx≤y≤1,
1≤x≤elogx≤y≤1,
which is the description we need for the new dydxdydx integration order. This latter pair of inequalites determine the bounds for integral.
We conclude that the integral∫10∫ey1f(x,y)dxdy∫01∫1eyf(x,y)dxdy with integration order reversed is
∫e1∫1logxf(x,y)dydx.