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Numerade Educator

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Problem 32 Medium Difficulty

Express the integral as a limit of sums. Then evaluate, using a computer algebra system to find both the sum and the limit.

$ \displaystyle \int^{10}_2 x^6 \, dx $

Answer

$\frac{9999872}{7}$

or $1428553.143$

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Video Transcript

to evaluate the Senate grill using limits and some So we could first find Delta X R B is our upper limit, which is 10 minus a are lower limit all over and and that would end up giving us eight over em. So that is dull tax. And then we'll go ahead and right out the limit and the some statement together. So it's going to be the limit as n approaches infinity of the some from high is one up to end of we'll go ahead and write the with Delta X here, which is eight over end and then multiplied by this X to the sixth. But to start that off, we actually need the lower limit is where we need to start, which is to plus Delta acts times I. So it's going to be plus eight hi over and all of that is then going to be to the sixth. So at this point, we could use a computer, um program toe. Evaluate this. Let me go ahead and highlight what I plugged in first. So then I plugged this end first, the some and that turned into basically, we're gonna go ahead and write out just the first term because only the highest power of n really matters here. The rest of the terms get knocked out, so it's point yes. 29 1234 and then 616 all divided by 21. And then that was times and to the sixth. And that was also divided by under the sixth. There's more terms here, but they have lower powers of N So, for instance, times end to the fifth. And then when you take the limit as an approaches infinity, these air all going to go to zero thes ends to the six will cancel. So our solution is really going to be this, and it turns into a couple things that are all the same. So another way to write that using approximation is 142 eight, 1428553 point one, and then it keeps going for three. And if you were to plug this into a new algebra system or basically like a graphing calculator and evaluate that, it'll end up giving you this exact value here. Well, that's approximate value here, I should say so. It's pretty cool how you can see the some and the limit do result in the correct value here as well