Problem 1 Medium Difficulty

Evaluate the Riemann sum for $ f(x) = x - 1 $, $ -6 \le x \le 4 $, with five subintervals, taking the sample points to be right endpoints. Explain, with the aid of a diagram, what the Riemann sum represents.

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04:37

Frank L.

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in order to evaluate the integral here with a three months. Um Then the first thing we're going to do is find out what Delta X is. B is four a is six negative negative turns that into a positive, all divided by and the number of sub intervals, which is five. So this turns into plus, so that's 10/5 or two. So we're going to use the heights determined by the right endpoints. So the first height that we're gonna actually use and thes we're all gonna have wits of two. So let me just go ahead and put those in here first. We're going to go by twos. In other words, Oh, this is clearly not drawn to scale. Okay, so we're gonna go down from four and then over, and that will be the first area second area determined by negative, too. Third area determined by zero. Then the next area determined by positive, too. And then finally, by positive or here, one way to write that says you could just factor to out. And then we're really adding half of negative four. Pull us, uh, both negative too. Plus F zero all that I'm doing in this formula is adding the with times the height. So the width is too, which is why that's factored out front. And the height of our first direct angle is f of negative four. You multiply those together and it gives you the area. And I'm just adding up all 12345 of these rectangle areas here. So the next one would be plus f of to and then finally plus f of four. Now, what's nice about the function that were given X minus one is we could actually just do this by hand without even using a calculator. Really? So we've got to Okay, half of negative four. Just plug in negative for here. So it's negative for minus one or negative. Five would be this first height here than f of negative two. Is this height for this rectangle here? That would be negative three. And then we have one more negative F zero. We plug in, we get minus one, Then we get f of to this height. I did not eclipse Go back. Not that height. This height here would only just be one. And then finally f of four and that would be plus three. So as you can see these simplify quite nicely. Negative three. Positive three Negative one positive one All cancel two times negative. Five. Therefore, the right Remond sons end up giving us a new estimated area of negative 10 for the area under the curve of why equals X minus one from negative six out to four. So I hope that was helpful there.