The reason why you get an answer that varies with x, rather than just an plain number, is that your limits on the outer integrals vary with x. The limits should be from 0 to 1 (and similar).
The question
Use double integration to find the area bounded by and
My attempt
I know that finding the area of a bounded region is given by:
So I drew the graph and decided to attempt it by taking two integrals:
+
But upon solving these I get rather than the solution
I'm fairly sure I'm attempting this wrong. :/
Any help would be most appreciated!
Pretend this isn't a double integral we're evaluating; instead, think of this as a problem from single variable calculus. Suppose we wanted to find the area between the curves and . What would you do first? Find the points of intersection! Clearly, they are the points (0,0) an (1,1).
If you integrate with respect to x (treating it as a type I region), then the integral for area would be , BUT...as a double integral, this is just simply .
Analogously, if you integrate with respect to y (treating it as a type II region), then the integral for area would be , BUT...as a double integral, this is just simply .
In this case, the first integral is easier to evaluate. I leave it for you to come up with the desired result.
I hope this makes sense.