Two young mathematicians discuss what calculus is all about.

Check out this dialogue between two calculus students (based on a true story):

Devyn
Ah. So now we have a connection between derivatives and integrals.
Riley
Right, the derivative of the accumulation function is the “inside” function.
Devyn
So how do we use this to compute area?

Sometimes it helps to think about the most basic examples. Consider

\[ \int _2^5 4 dt \]

We know (by geometry) that this computes the area of a \(3\times 4\) rectangle which equals \(12\). On the other hand, if we consider the accumulation function

\[ F(x) = \int _2^x 4 dt, \]

we see that

\[ F(5) = \int _2^5 4 dt. \]
What is \(F(2)\)?
\[ F(2) = \answer {0} \]
On the other hand, the First Fundamental Theorem of Calculus says that if
\[ F(x) = \int _2^x 4 dt, \]
then \(F'(x) = 4\). Armed with this knowledge, and the fact that \(F(2) = 0\), what must \(F(x)\) be?
\[ F(x) = \answer {4x-8} \]
2025-01-06 19:50:42