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Use Definition 2 to find an expression for the area under the graph of $ f $ as a limit. Do not evaluate the limit.

$ f(x) = \sqrt{\sin x}, \hspace{5mm} 0 \le x \le \pi $

$\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \sqrt{\sin \left(\frac{\pi i}{n}\right)} \cdot \frac{\pi}{n}$

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Campbell University

Harvey Mudd College

University of Nottingham

Idaho State University

for this problem. We've been given a function f of X equals the square root of the sine of X, and we're going to be looking at the X is between zero and pi inclusive. The goal for this problem is to use definition to to write out the area. But the area is under this curve as a limit, and we're not going to evaluate the limit. We just need to get it set up. So let's review what definition to says. It says that the area of a region under a graph is the limit of the some of the areas of the approximating rectangles. In other words, as I get more and more and more rectangles and there's a number of rectangles I have said, let that go to infinity. I'm gonna be summing up all of the rectangles. So rectangles, that's a height and the base so the height gets me up to the function. So it's F of X sub one for my first rectangle times The base, which is Delta X, the second rectangle f of X up two times the base, and it's the same base Delta X F of X, up three Delta X and so on, all the way through the 10th rectangle times the base. Okay, so let's think about how this is going to work. Each one of these rectangles has that base and a height. So I can think of this using summation notation, and it's going to infinity. I'm going to have I need to have some, uh, some variable that I'm indexing through. So when I do my summation notation so I'm going to say X I equals one to end. So I was gonna start with one go to end. Our end is getting bigger and bigger and bigger, so ugly, summing up the base times the height. So let's think about what those base and heights look like in terms of this particular, um, function. Okay, first the base. Well, the base is Delta X. In each case, what is Delta X? Well, the entire distance along the bottom of this function is pie. I'm going from zero to pi, so I have pie units across the entire base. I'm breaking that pie units up into end rectangles so that Delta X is pi over n. Okay, there's my base now. What about the height. Well, the height is the function at a point within that interval. So I know it's going to be the square root of the sign of something. But how can I reference? How can I explain that here with some of the letters already have, like I and and and things like that. Well, where am I on that distance from zero to pi. If you imagine each one of these I'm breaking this up into intervals are getting smaller and smaller. This interval is pi over end. So if this gets very, very, very small, I'm going to be picking a number in an incredibly small area. So I can say that I'm going to pick pi over end. Okay, So as I go to the second unit, it's gonna be pi over n times to that gets me over to my next bit. When I go to my third unit, I can say pi over n times three and that keeps putting me to the next to unit to the next to the next to the next. As I walk across this distance so I can say this is sine of pi times I over end. So that is using this definition to write the area as a limit as n goes to infinity

Rochester Institute of Technology