"Накопичення" функції

Suppose we want to find the "total" of the quantity $y=2x$ as $x$ ranges from $3$ to $4$.

Instead of thinking it as taking a sum $$2(3)+2(3.0001)+2(3.0002)+\dots+2(4),$$ we try to find the area under the line $y=2x$. (Does this make sense?)

We can find the area under the line by integrating $y=2x$. We will use integration of monomials from Integral Calculus, but you could also use the triangle area formula from Geometry.

This is because the derivative of $x^2+C$ is $2x$.

Anyway... we want the area of the curve between 3 and 4. Let $f(x)=x^2+C$.

Then we want $f(4)-f(3) = 7$. Yay! We got the answer.

We see the anti-derivative of $y=3x^2$ is $y=x^3+C$. Then let $f(x)=x^3+C$. We have $f(5)-f(3)=98$