I came across this proof that the definite integral $\displaystyle \int x^k \, dx=\frac{x^k+1}{k+1}$. The proof sums up many rectangles which are below the the curve (the lower sum), with the limit that the number of rectangles tends to infinte.
https://plus.maths.org/content/intriguing-integrals-part-i
Please have a look at the link.
I was wondering if this proof was valid as it only considers the lower sum. I attempted to prove it for the upper sum, but did not come out with the same expression.
it depends in the definition of integral you have. I refer to the book Principles of Mathematical Analysis of W. Rudin.
In my words... We define the Lower integral as the "limit" of the lower sums, and the Upper integral as the "limit" of the upper sums. When that limits are equal we say that the Definite integral corresponds to that limit. I put quotation marks because I'm not very strict in say just limit.
In this case I think is valid to only prove with the lower limit because the function satisfy the conditions for be integrable; but if we want to be more strict (and if, in hypothesis, you don't know the conditions under a function is integrable) we need to prove that the limit of the upper sums is equal to the limit of the lower sums.