We can consider that a rectangle that is a bit bended. And the hieght is 27, but the base is not sqrt 2 and you need to calculate it.
But, to clarify this to you, I will change the example:
What if you have a fence of height 27 over the unit circle? What would be its area? Do you think that the area is 0?
What if you have the same but only from 0 to pi? Do you think that the area is now
(1-(-1)) x 27?
This instgeral is NOT path independent. You actually need to calculate it.
If you still don't see it, do a drawing and think of you fencing your yard. Forget about integrals, pretend you only attended the primary school. How would you alculate the amount of paint in gallons for this fence?