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