Multiple Integrals for Functions Unbounded at Isolated Points

[silvershadow7]

In a recent homework assignment, I was asked to prodive a definition for ∫f(x) in the Region D, provided there was a discontinuity somewhere in the region. To define the integral, we merely removed a sphere centered on the discontinuity of radius δ>0 and found the limit of the integral as δ→0.

My question is how would you provide a more generalized definition for a function that had multiple undefined points? if i had points (x1,y1) and (x2,y2) where the function was undefined.

Would I somehow split the integral up into three parts?

 

[mathman]

Whatever trick you used can be used separately at each point in question. You would need a more sophisticated approach if there are an infinite number of points involved.

 

[silvershadow7]

but if the integral originally broke up the bounds to be one integral with (a, δ) and the second integal with(δ,b), with δ going to zero after the integral was evaluated, then how would the bounds look with more points??

(a,δ1) , (δ1,δ2), (δ2,δ3), ...etc (δn, b) ??

with each of those δ, going to zero??

 

[mathman]

Your notation is very confusing. The general idea is to treat each discontinuity separately.

You use a, b, various δ's, without defining anything.