32
$\begingroup$

I know that derivative is the slope of the tangent line, and that integral is the area under the curve.

My question is that how these two distinct concepts are geometrically related? What is the relation between the slope of the tangent line and the area under the curve?

If these are inverse of each other, then there should be a relation between them, I believe.

$\endgroup$
72
$\begingroup$

The magic word is "rate of change". The slope is the rate of change, so the slope is simply "how much the function grows when you move right". So if you plot the integral curve (which geometrically can be interpreted as an area under some other curve), then the rate of change is "how much the area grows when you expand it to the right", which is exactly the value of the original function.

Sketch:

On the left, you have a curve of the original function $f(x)$, and the approximate area under it, shaded as red rectangles. The curve of the integral $\int f(x)dx$ is obtained by "adding" the areas together (right figure, the $y$ coordinate measures the total area of the curve in the left figure). The derivative of the right curve is the slope (dashed lines across the rectangles), and obviously, the slope of a rectangle with unit width is exactly its height, and the height of the rectangle brings you back to the left figure and the value $f(x)$.

function integral

$\endgroup$
  • 7
    $\begingroup$ it's a pity I can only upvote this once. Good job. $\endgroup$ – hjhjhj57 Jul 21 '15 at 7:07
  • 4
    $\begingroup$ I wish I had seen this example when learning calculus! $\endgroup$ – FundThmCalculus Jul 21 '15 at 16:36
  • 6
    $\begingroup$ ... and that's the Fundamental Theorem of Calculus! Beautifully explained. $\endgroup$ – Simon Kuang Jul 21 '15 at 17:44
  • 2
    $\begingroup$ @orion, Thank you very much. I want to reproduce such diagrams for a few functions. May I ask how did you draw the diagrams? Which software you have used? $\endgroup$ – Javad Kouhi Jul 28 '15 at 19:35
  • 2
    $\begingroup$ @JavadKouhi I used asymptote (asymptote.sourceforge.net). It's not just for plotting, it's a fully qualified programming language, which is why it's very easy to do whatever you want. It's perfect for geometry sketches. The examples are stunning: piprime.fr/asymptote $\endgroup$ – orion Aug 2 '15 at 9:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.