# Interchanging integration bound for double integral

1. Feb 7, 2005

### hytuoc

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 ?

2. Feb 7, 2005

### dextercioby

What's the use,u'd still have to integrate first after "y" and then after "x".You can rewrite it

$$\int_{\frac{1}{2}}^{1} dx \int_{x^{3}}^{x} f(x,y) dy$$

It's the most elegant & lucrative way.

Daniel.

3. Feb 8, 2005

### Galileo

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.