Changing bounds of integration.

[Kuma]

Homework Statement

The problem given:

Homework Equations

The Attempt at a Solution

I need an x^2 in there to do the inner integral. I'm having a bit of trouble figuring out how the bounds are defined. X goes from 1 to y/2 and y goes from 0 to 2. So does that mean x goes from 1 to 2x?

 

[vela]

Draw a sketch of the region you're integrating over in the xy plane.

 

[Kuma]

I did that but I'm unsure if its right.

It should just look like a triangle from my drawing. Y going from 0 to 2 and x going from 2x to 1

So then the changed order should be

y going from 0 to 2x and x going from 0 to 1?

 

[vela]

I did that but I'm unsure if its right.

It should just look like a triangle from my drawing. Y going from 0 to 2 and x going from 2x to 1

Yes, but you're saying it in kind of a confusing way. When you say "x going from 2x to 1," you're using "x" to mean two different things. It would be better to say x goes from y/2 to 1.

So then the changed order should be

y going from 0 to 2x and x going from 0 to 1?

Yes, that's right. The triangle is bounded on two sides by the x-axis and the line x=1. The third side is the line y=2x or, equivalently, x=y/2. The first form is useful when you integrate with respect to y first; the second form is useful when you integrate with respect to x first.