integral question would like help understanding it?

Question

If you have an integral of some function (f(x) where F'(x)=f(x)) over interval [a,b] where a=b then aren't you not taking the integral? Isn't that just like what the fucnction should be? Isn't it just the line segmint at a to F(a)? So I am unsure whether or not the integral would be zero. You have no width on your area but you have height (excluding F(a)=0). Am I confusing the idea of the added constant with the integral itself or am I just really confused? Is that integral always zero because your height is one (or zero) dimentional, and the integral is a 2-D quantity?

Answer 1

The integral represents area, and lines are infinitely thin and thus have no area. This is why the integral is 0.