The antiderivative at x is defined as the area under the curve from 0 to x, which the Riemann sum gives a nice intuition for how you can get from the derivative.

So to get the area under the curve between a and b, you calculate the area under the curve from 0 to b (antiderivative at b) and subtract the area under the curve from 0 to a (antiderivative at a).

At least that's my sleep deprived take.

I took calculus in high school and college, and I don't think any of my instructors explained the intuition as well. So sleep-deprived or not, it's a great one!