Interchanging integration bound for double integral

In summary, to interchange the integration bound for the given function, you can rewrite it as the integral from 1/2 to 1 of the integral from x^3 to x of f(x,y) dy dx. However, if the bounds are independent of x or y, the order can be exchanged only if f is continuous on the domain of integration. If one of the bounds depends on x, it may not be possible. In this case, it is possible but the integral must be split into two. To do this, draw the region bounded by x=1/2, y=x, and y=x^3 and write the integral over this region by first integrating with respect to x. This is considered the most elegant and
  • #1
hytuoc
26
0
How do I interchange the integration bound for the function below (change to dx dy):

Integral from 1/2 to 1, integral from x^3 to x [f(x,y)] dy dx ?
 
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  • #2
What's the use,u'd still have to integrate first after "y" and then after "x".You can rewrite it

[tex] \int_{\frac{1}{2}}^{1} dx \int_{x^{3}}^{x} f(x,y) dy [/tex]

It's the most elegant & lucrative way.

Daniel.
 
  • #3
If the bounds are independent of x or y, then the order can be exchanged if f is continuous on the domain of integration.
If one of the bounds depends on x, then it's generally not possible.

In this case it is, but you have to split the integral into two.
Draw the region over which you are integrating. It's the region bounded by x=1/2, y=x and y=x^3.
Try to write the integral over this region where you first integrate wrt x.
 

1. What is interchanging integration bound for double integral?

Interchanging integration bounds for double integrals is a technique used in mathematics to evaluate double integrals by switching the order of integration. This can be useful in simplifying the integrand or making the integral easier to solve.

2. When can interchanging integration bound for double integral be used?

Interchanging integration bounds can only be used if the integral is absolutely convergent. This means that the integral must converge regardless of the order of integration.

3. What is the process for interchanging integration bound for double integral?

The process for interchanging integration bounds is to first draw a graph of the region of integration and determine the bounds for each variable. Then, rewrite the integral with the new order of integration and solve as usual. Finally, check the solution to ensure that it is equivalent to the original integral.

4. What are the benefits of interchanging integration bound for double integral?

Interchanging integration bounds can make solving certain double integrals easier and more efficient. It can also help to simplify the integrand and make it easier to evaluate. Additionally, it can provide a different perspective on the integral and may lead to new insights or approaches for solving it.

5. Are there any limitations to interchanging integration bound for double integral?

Yes, there are limitations to interchanging integration bounds for double integrals. As mentioned earlier, the integral must be absolutely convergent in order for this technique to be used. Additionally, it may not always be possible to interchange the bounds or it may result in a more complicated integral. It is important to always check the solution to ensure it is equivalent to the original integral.

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