So I've been a calculus student now for about two years, and I've gone as high as differential equations, but I am still a bit puzzled by the fact that the area under the curve of some function is defined the way it is.
Let me clarify. so let's say you take the anti derivative of
$$
\int (x^2+4x) \ dx
= \frac{x^3}{3}+2x^2
$$
I don't have a clear intuition of why the area is exponentially greater than the original function. if any one could explain why that is, I would be very grateful. I know that this must sound like a stupid question, especially form someone who's been in math for quite some time, but I just don't have a good grasp on that concept.
Best Answer
The function $\frac{x^3}{3}+2x^2$ does not grow exponentially faster than $x^2+4x$. The function $\frac{x^3}{3}+2x^2$ is a polynomial, and it grows roughly like $(x^2+4x)^{3/2}$. That is not dramatically faster than $x^2+4x$. The following is an example of faster relative growth that is more familiar.
Look at the area of the region under the curve $y=x$, above the $x$-axis, from $x=0$ to $x=w$. If you draw the "curve" $y=x$, you will see that the region is a triangle with base $w$ and height $w$, so the region has area $w^2/2$. (The same result can be obtained by integration, but that's overkill.)
So the function grows like $w$, and the area grows like $w^2$, more precisely like a constant times $w^2$.
The most "extreme" example is a constant function like $f(x)=1$. The area from $0$ to $w$ under the curve is $w$. This grows much faster than $f$, which does not grow at all, but the rate of growth of the area would not be considered fast. And the fact that the area of a rectangle of constant height grows linearly with the base is unsurprising.
In general, imagine a curve $y=f(x)$, where $f(x)$ is positive, and, for simplicity, increasing. Let $W$ be the region below $y=f(x)$, above the $x$-axis, from $x=0$ to $x=w$. Then the region $W$ is contained in the rectangle with base the interval from $0$ to $w$, and height $f(w)$. So the area of $W$ is $\le wf(w)$. Thus, if $f(w)$ grows "fast," then the rate of growth of the area is not much faster than the rate of growth of $f(w)$.