# Showing why an integral is the area under a curve...

by Jd0g33
Tags: curve, integral, showing
 P: 326 So I've been spending a lot of time lately trying to figure out why an integral will give you the area under the curve. I asked the forum and got some great answers, but all were in terms of infinite sums, dx, and infinite rectangles. I think I've come upon a more fundamental answer that I haven't heard from anybody yet (although I'm sure its obvious to most people.) :) $y(x)$ is a curve on a graph. Now as $y(x)$ gets larger, the area under the curve $A(x)$ gets larger. In fact, say $y(x)=3$. Then the area at $x=1$ would be 3. Its rate of change at that instant would be 3. Which means the rate of change of the area under the curve is equal to y(x): $A'(x)=y(x)$. That being said, the actual area under the curve would equal the anti-deriviative of y(x): $A(x)=∫y(x)$. Is this correct? If so, why do we use the confusing dx when it's really not even a number that you could multiply by to get the area of a rectangle (other than showing the integral is with respect to x). I guess I just don't like the $\frac{dy}{dx}$ notation. Thanks