Definite Integral of an Odd Fractional Power, various limits: Find .

First we note that this is an odd function, so that there will be as much negative area as positive area for these symmetrical limits. Hence, we expect the result to be zero.

Next we apply the Fundamental Theorem of Calculus:

, as expected.

Now try 0 and 1 as limits. Since the function is positive in this interval, we expect to get a positive area: , as expected.

Now try -1 and 0 as limits. Since the function is negative in this interval, we expect to get a negative area: , as expected.