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

Think of it like we’re drawing a mountain. You place your pencil on the paper, and you move it left to right.

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The higher up you go, the more “massive” the mountain is that you’re drawing. Moving the straight across increases the mass of the mountain. However, if instead you move the pencil upwards and you move to the right, the mountain is getting even bigger, right?

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And if you move the pencil diagonally down, that would make the mountain less massive than if you had moved straight across.

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This is the intuition that shows the relationship between the slope of a curve, and the area underneath it. the “mass” of the “mountain” is the area.