Why Does Integration of Certain Functions Result in Infinity?

[touqra]

Using Maple 8, I integrate this:

$$\int_0^1\frac{1}{\sqrt(p(1-p))} dp$$

and I get $$\pi$$
but, this function goes to infinity at p = 0 and 1.
How can this be possible ?

 

[Hurkyl]

How can this be possible ?

Why wouldn't it be?

 

[touqra]

Why wouldn't it be?

The denominator has sqrt[p*(1-p)], which will give infinity on both p = 0 and 1. And so, the area under this curve is infinity too, right ?

 

[Hurkyl]

Can you think of a reason why that should be true, though? If you're convinced the area should be infinite, then try to work out a proof of it -- such exercises are often really good at clearing up misunderstandings.

Incidentally, it might help to play with simpler functions, e.g.

$$\int_0^1 \frac{1}{\sqrt{x}} \, dx$$

or

$$\int_1^\infty \frac{1}{y^2} \, dy$$