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Problem 30 Easy Difficulty
Express the integral as a limit of Riemann sums. Do not evaluate the limit.
$ \displaystyle \int^5_2 \biggl( x^2 + \frac{1}{x} \biggr) \, dx $
Answer
$\lim _{n \rightarrow \infty} \frac{3}{n} \sum_{i=1}^{n}\left(2+\frac{3 i}{n}\right)^{2}+\frac{1}{\left(2+\frac{3 i}{n}\right)}$
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Video Transcript
getting this integral as a limit of sums. We could go ahead and start off by putting the limit as our variable M approaches infinity. And then we're going to some from eyes one up to end. Now, the first thing that we calculate, we look at two and five and then we go ahead and do be minus a over and basically get like the width of each of our rectangles. B is five minus. A is to over end and this turns into three over end and so we could go ahead and put that out front of the sun there. And then the next thing we do is we start from the lower limit and then we add that amount with its i f component. And so from there we could go ahead and write that next part. As to which is just like we said, our lower limit, the starting amount Waas. This helps us find the width of the the, uh, rectangle here three i over. And now all of that is basically this and so that's why it needs to be squared. And then we have plus one over that same amount and so once you find what X is replaced with in one area the same thing in the next. Let's make sure we leave enough room there, so that'll be three I over and and then, lastly, the DX is replaced with a Delta X there. So basically, that's the idea except Delta X. We've already kind of written as this, and so that would be our final limit in this case here.
University of Utah
Top Calculus 1 / AB Educators
Anna Marie V.
Campbell University
Heather Z.
Oregon State University
Kayleah T.
Harvey Mudd College
Joseph L.
Boston College