Is the Limit Definition of a Definite Integral Correct?

[GreenPrint]

\int^{b}_{a} f(x) dx = lim_{c → a^{+}} lim_{d → b^{-}} \int^{d}_{c} f(x) dx

Is this true?

 

[HACR]

\int^{b}_{a} f(x) dx = lim_{c → a^{+}} lim_{d → b^{-}} \int^{d}_{c} f(x) dx

Is this true?

What are the arguments of the limit in this case?

 

[GreenPrint]

What are the arguments of the limit in this case?

a and b on the left hand side and I replaced these with c and d evaluated with limits with a and b, a from the right and b from the left.

 

[maCrobo]

Yes it always true, either if f(x) is continuos on [a,b] or on (a,b). In the first case you have a proper integral, in fact the primitive F(x) is also continuos and the limit is the value of F at a (or b). In the second case you have an improper integral and in that case is necessary to use the limit to check whether or not the result is a finite value.

 

[GreenPrint]

hmmm...
ya i guess so thanks