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 used to drive me crazy in high school that the derivative of the volume of a sphere (four thirds pi r cubed) is the surface area of a sphere (four pi r squared). No teacher ever addressed this “coincidence.”

But after thinking about it for awhile I realized that if a sphere was growing, the tiniest little instantaneous additional volume being added would be the size of its surface.

That helped me a lot. Maybe it helps you too.