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 a function as just the change in y. If it goes from 2 to 5, y is 3. Now, the slope is y amount (2 to 5) divided by the x amount it took to change from 2 to 5. If x changed from 0 to 6, that’s 6 for x, so the division (the derivative) is 0.5. For each x, y increases in 0.5.

Now let’s do the opposite. Instead of divide y by x, multiply it. What is the y amount multiplied by the x amount in our example? 18. Ok, good. Now what is x (width) multiplied by y (height). It’s the area!