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Numerade Educator

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Problem 31 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^{\pi}_0 \sin 5x \, dx $

Answer

$\frac{\pi}{2 n} \left(\frac{\sin \left(\frac{\pi}{2 h}(a n+5)\right)+\cos \left(\frac{5 \pi}{2 n}\right)}{\sin \left(\frac{5 \pi}{2 n}\right)}\right)$

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Video Transcript

So problem. 31, all we know about definite integral so far is the limit definition. So this is the limit as in approaches infinity. Mhm. Of the sum of all the areas of the rectangles. So if I have in rectangles that's going to be I equal one to end. The width of each rectangle is Pi zero over in. And then you're gonna have the sign Of and it's gonna be the height that each of these rectangles. So that is going to be with the with five power in. So sign of five pie I over in. And that is it. So now it asks us to use our computer outdoor system to evaluate this one because it is quite complicated. So if I go over here the sum and you'll see that the sum of um I'm equal one to end of pira ensign five pi over in its given by this complex expression. So the some which you see right here. So this is going to be the limit as N approaches infinity. And then that some is just simply will not simply. But it is pi times the coast secret of five pi over to in times the sign a pie. Nine in plus five over to in plus the co sign of five pie or two in. Mhm. Yeah. All of that divided by two So. so quite complicated. Now they say go back to your computer algebra system and evaluate the limit. So there's are some to evaluate the limit. We just simply say what is the limit as in approaches. Thank you infinity. Uh huh. Yeah. and the answer there is 2/5. So this is 2/5. This problem becomes a whole lot easier once we move past the limit definition and you have rules with the fundamental theorem of calculus to calculate this.