Improper Integrals

How do we find the area of on and on ?

Improper Integral

Definition

If is continuous on , then the improper integral is defined as:

If this limit exists, then we say that the integral converges. Otherwise, it diverges.

Also,

In essence, we're just replacing the with which approaches infinity.

Interesting note, is that we are looking at a region with infinite length, but finite area.

Definition

If is continuous at every point on EXCEPT at , then the improper integral is defined as:

Similarly, if the discontinuity is at instead, then

And if the discontinuity is at point , then

Example

Find the area of on

Example - Diverging Integral

Find the area of on

The integral diverges.

Example - Calculating Escape Velocity

The kinetic energy must be greater than the work done. Since the force of gravity varies with distance, we have:

We also have the universal gravitation equation:

We want to move the projectile from to .

So to escape Earth's gravitational field, we need: