Why is the Integral an Anti-derivative?
We will use the following definitions:
Anti-Derivative : The anti-derivative of a function is a function such that .
Integral (area definition) : The integral of of is the area under the curve from to .
Let us define the area under the curve from some starting point (it’s arbitrary) to as . Now we calculate . It is left to the reader to show:
Dividing by and taking the limit approaches zero we get.
Thus, the integral of is the anti-derivative of .
By drawing a sample curve of and using the formula for the area of a trapezoid, show that:
Hint: For small enough, the path from to is a line.
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