Family of Curves: Writing an Integral as a Summation

[lilcoley23@ho]

f I consider the area of the family of curves as y = (1 - x^1/p)^n where x is greater than or equal to zero but less than or equal to one, I can write that in as integral as

the integral from 0 to 1 of (1 - x^1/p)^n dx but I'm not sure how to write that as a summation, which I have been trying to do. I know the summation looks like:

summation from k=0 to n ((-1)^k (n k) p/(p+k))

Can you please help me with this?

 

[mathman]

I am confused (although it looks like to me you are confused) as to what your summation is supposed to represent.

First you talk about a family of curves. What is the parameter defining the family (n or p)?

The you talk about an integral over x, but p and n seem to be fixed.

Finally what are you using the sum for? Are you representing the integral as a sum? It doesn't look like that at all.

 

[lilcoley23@ho]

My summation as well as the integral is supposed to represent the area under the curve.

n and p are just positive integers. I need to figure out how the area under the curve can be written as the integral equal to the summation.

 

[HallsofIvy]

The area under what curve? You have, as you said, a family of curves and each curve in that family has a different area. Certainly, summing over n will NOT give you the area under any curve.