if mathematically derivatives are the opposite of integrals, conceptually how is the area under a curve opposite to the slope of a tangent line?

Question

if mathematically derivatives are the opposite of integrals, conceptually how is the area under a curve opposite to the slope of a tangent line?

In: 332

Answer 1

It’s probably easiest to break it down in terms of simple geometry.

Let’s draw two points. Now, write the equation for a slope between those two points:
(Horizontal Distance Between Points) / (Vertical Distance Between Points)

Now, write the equation for a rectangle that has those two points at opposite corners:
(Horizontal Distance Between Points) * (Vertical Distance Between Points)

In other words, the first (the derivative) is X / Y while the second (the integral) is X * Y. Does this make it easier to understand why they’re inverse operations?