# Thread: Prove that a function is integrable

1. ## Prove that a function is integrable

Hey all,
I am supposed to solve problems like these. If someone can help me with this, I should be able to figure out the rest. I just dont know which theorem to use out of all the given ones.

PS: Its the first time I'm using latex to submit a question, I was able to figure out how to adjust the image sizes, but not the spaces between words. Apologies.

$\displaystyle Prove that \int_0^b x^3 dx = b^4/4, by considering partitions into n equal$
$\displaystyle subintervals, using the formula \sum_{i=1}^n i^3 = n^4/4 + n^3/2 + n^2/4$

2. Well, what have you tried to do yourself?

If you divide the interval from 0 to b into n equal length subintervals, what is the length of each subinterval? If you take $\displaystyle x_i$ to be the right hand endpoint of each subinterval, what is $\displaystyle x_i$ in terms of i? Then $\displaystyle (x_i)^3$ is the height of each rectangle what the length of the subinterval is the base. The area of each rectangle is the product of those two. Find the entire area by summing those. With any luck, you will get a sum that looks like $\displaystyle \sum i^3$, perhaps multiplied by other numbers.