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Problem 2 Easy Difficulty

If $$ f(x) = \cos x \hspace{10mm} 0 \le x \le 3\pi/4 $$ evaluate the Riemann sum with $ n = 6 $, taking the sample points to be left endpoints. (Give your answer correct to six decimal places.) What does the Riemann sum represent? Illustrate with a diagram.

Answer

Riemann Sum $\approx 1.033186$

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Video Transcript

All right. We've got a question here. The function of X. You get to co sign x Where X is anywhere for 1 To be forward spy. He was a hero. 3/4 fine. We want to calculate agreement some And and is equal to 6 from the left end points. Alright. And where we calculate this Is we're going to take our series jeans. The left hand points. The same thing as writing and minus one R equals to 1 fc by Multiplied by the change and apps. All right. The change in Mexican calculated by taking B -A over N. In this case your baby is going to be your larger X value just 3/4 pie and your is your smaller X value zero. And that's going to come out to be pi over 6. Susan Pirate eight. That's your change in X. And then the way you calculate the left end points, you start off by 0 And then you just continue to add pi over 8 until you get two x minus one and minus one. Which would be six terms minus one which is five terms You go to x sub 5. All right. Finally we put that together in the series. So our series would then be changing. It's like you said here, Which is Pi over 8 Multiplied by a function of X 1 0 cost function of that's up to Which is ascending this pie over 8. What's the function of? That's So to which is to Pile 8. And you just keep going on until you get to the function X supply which is far All right. When we go through and actually solve for those what you're gonna do. Just plug in those x values here into your function of X and solve for the function of X one X zero function excess by or eight function Excellent. Access all the way to five pi or eight. You add them all up and multiplied by your higher eight. And when you do that you'll be able to get to your final answer which comes out to be approximately 1.033186. All right. That's our final answer there. And then it also wants us to show this with representation of a graph. All you're basically doing is you're having your ex values here Function of that year. And so you would have It be a larger value in excess pile of eight And it progressively gets smaller Until finally you get to a point where we know that whenever we cross that pirate half boundary our coastline will become negative. And so you have Something that starts to look like this. All right. And these are just your representation. So that's one X two X. Dx 456 Excuse me, ex Except 0 to accept 5. And then if we wanted to draw a line, we have something that looks like this And that will be kind of a representation in graphic uh in a graph form. And regardless of all that, though, your final answer is going to be this right here, hope that clarifies the question. Thank you so much for watching.