Why is the Integral an Anti-derivative?

Calculus Tutorial

Why is the Integral an Anti-derivative?


We will use the following definitions:

Anti-Derivative : The anti-derivative of a function f is a function F such that F' = f.

Integral (area definition) : The integral of \int_a^b f(x) dx of f is the area under the curve f(x) from x=a to x=b.

Let us define the area under the curve f from some starting point (it’s arbitrary) to x as A(x). Now we calculate A(x+dx) - A(x). It is left to the reader to show:

(1)   \begin{equation*} A(x+dx) - A(x) = \frac{dx}{2} \left( f(x+dx) + f(x) \right). \end{equation*}

Dividing by dx and taking the limit dx approaches zero we get.

(2)   \begin{equation*} A'(x) = f(x). \end{equation*}

Thus, the integral of f(x) is the anti-derivative of f(x).

Sample Problem

By drawing a sample curve of f(x) and using the formula for the area of a trapezoid, show that:

(1)   \begin{equation*} A(x+dx) - A(x) = \frac{dx}{2} \left( f(x+dx) + f(x) \right). \end{equation*}


Hint: For dx small enough, the path from f(x) to f(x+dx) is a line.

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