Contents of this Calculus 1 episode:

Definite integration, Area under the curve of a function, Integrable, Primitive function, Newton-Leibniz formula, Fundamental Theorem of Calculus, Area of region that bounded two functions.

Here is a function,

to which we draw a tangent at x=3.

This way we get two regions.

One is bounded by the function, the tangent and the y axis,

the other one is bounded by the function, the tangent and the x axis.

We need to find the area of these regions.

Well, we will likely need the equation of the tangent.

Well, aren't we lucky? Here it comes:

Now let's get down to business.

It is much easier to calculate the area that is bounded by the y axis.

Since this is a normal region, it is sufficient to integrate the difference of the two functions:

Calculating the other area will be a lot more unpleasant.

First, we will need these intersections.

And now let's see the areas.

The area in question:

Calculus 1 episode