Contents of this Calculus 2 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.

Definite integration deals with calculating the area under the curve of a function.

Here is a function, and the area under its curve from a to b is:

This is assuming, of course, that f(x) is integrable on the closed interval [a, b], and it has a primitive function on this interval.

This primitive function is called F(x): the indefinite integral.

If such primitive function does not exist, then calculating the area under the curve turns into a nightmare.

Nightmares will be discussed in a separate slideshow.

Instead, let's try how this theorem works, and find the area under the curve of x2 from 0 to 1.

According to Newton and Leibniz, this area is:

Here comes the primitive function:

And this is where we have to substitute first 1, and then 0.

Let's see how we could make all this a bit more exciting.

For example, let’s find the area of the region that is bounded by functions f and g.

Here is the plan:

First, we calculate the area under the red function from a to b,

then we do the same for the yellow function,

and finally, we subtract them.

At the same time, it would not hurt to know what a and b are.

Well, the special thing about a and b is that the two functions are equal there.

So, we have to solve this equation.

These type of regions of which we calculated the area are called normal regions.

A normal region is bounded both above and below by a function,

and on the sides by lines x=a and x=b.

Sometimes the two functions meet on one side,

but they can meet on both sides, too.

The area of a normal region is:

or, if function g happens to be on top, like here in our drawing,

then the other way around.

The advantage of this method is that we only have to integrate once. Let’s see it.

For example, let’s find the area of the region that is bounded by functions f and g.

First we calculate the intersections,

Calculus 2 episode