How do I know which region to use?

Question

I'm told to use double integrals to calculate the area of the region bounded by

y = 2x
x = 0
y = 1 - 2x - x²

I know how to set up and calculate the double integral, so I don't need help with that.

However, when I graph this, I see two different regions that are bounded by these lines, one region to the left of x = 0 and one to the right of x = 0. Am I seeing this correctly?

(I'm going to calculate the area for both regions, unless someone can prove to me that there is only one region with these bounds.)

I've given all the information that I have, which is why I posted my question.

I think all I can do is calculate the areas for the two different areas. I'm just trying to see if anyone else agrees or disagrees (and why).

Answer 1

There are two different regions, as you say. Is there something in your problem statement that would allow you to eliminate the area where x<0?

Some problems have an implicit limit on the integral because it is a physical problem. Other problems may explicitly say integrate from x=0 to x=infinity (or the other limit).

There must be a way to discriminate between the two regions; otherwise, the x=0 contraint would be irrelevant.

Answer 2

more info plz