## Calculus Tutorial

*Why is the Integral an Anti-derivative?*

#### Intro

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:

(1)

Dividing by and taking the limit approaches zero we get.

(2)

Thus, the integral of is the anti-derivative of .

#### Sample Problem

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

(1)

#### Solution

Hint: For small enough, the path from to is a line.

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Math And Physics From Algebra To College Calculus |

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