Problem 4 Medium Difficulty

(a) Estimate the area under the graph of $ f(x) = \sin x $ from $ x = 0 $ to $ x = \pi/2 $ using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate?

(b) Repeat part (a) using left endpoints.

Answer

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Video Transcript

so estimate the area of the graph for zero to pi over two graphics therefore work civil science so destroyed very quickly. So I would only draw from zero to pi over two parts. So as pie over X equals pi over two, sex is one and X equals zero. Science zero. So if we're using 4 10 goals, that means, um we have value pi over eight pi over four and we caused the interval into Sorry. This Yeah, it should be three pi over eight. If we are using our four approximating are rectangles, then then the area should be the extract for a tractor. If we use right and point, then the rectangle looks like this. Four seconds can go. Use pi over four For the third rectangle, use 8/4. The fourth rectangle. We use pi over two and the area estimate should be the base of each rectangle, which is pi over eight times the sound of height. So Cy Pi over eight plus sign pi over four plus by three pi over eight plus sign pi over two. So this is right and point and this will be overestimated so you can see from the picture. Because if you are estimating increasing function, you use the right and point it will be the the largest, the largest value in that interval. And if you use the left hand point will be the smallest value. Therefore, uh, underestimate. So let me write down the formula Four left and point. Sorry. So the area is still the base of the rectangles times the sum of high in this case instead of starting at eight pi over a start at zero. Oh, plus sigh. Okay. Sorry. This should be three pi over eight. Yeah. So you know you're left with, uh, this will be underestimated, by the way. And some of you may not know the value of the trigonometry functions like signed pi over eight size three pi over eight. So here is some identity that can help you. For example, we know that home. Mm hmm. Oh, coz I two things to equals one minus two side square set up. So if you put instead, I equals pi over eight, then two things I will be pi over four. So 11 side will be square root of 2/2, because what minus two side square. Hi over eight. So from here we can solve side pi over eight. So, for example, this one, By solving this equation, we get this e course one minus square root of two over to divided by two. And because sign pile, raise positive. So here we can just take the square root without putting the negative side. And that's the value of signed by over eight. And you can figure out science three pi over. It's a similar way.