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Estimate the area between the graph of the function $f$ and the interval $[a, b] .$ Use an approximation scheme with $n$ rectangles similar to our treatment of $f(x)=x^{2}$ in this section. If your calculating utility will perform automatic summations, estimate the specified area using $n=10,50,$ and 100 rectangles. Otherwise, estimate this area using $n=2,5,$ and 10 rectangles.$$f(x)=\sin x ;[a, b]=[0, \pi]$$

1.57,$1.93 ; 1.98$

Calculus 1 / AB

Calculus 2 / BC

Chapter 5

INTEGRATION

Section 1

An Overview of the Area Problem

Functions

Limits

Differentiation

Integrals

Integration

Integration Techniques

Continuous Functions

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So in this problem, we're working on a different interval. So things are gonna look a little bit different for this problem. We're dealing with the function F of X is equal to the sign of X And we're on the interval from zero to pi So we're gonna count The different number of rectangles are gonna do to rectangles five rectangles and 10 rectangles The more rectangles I use, the closer my approximation will be The first thing I want to do is we want to add up all of the heights together and then multiplied by the with. So we're gonna do here is we're going to take the width of the of the intervals. So if there's pie intervals, we're gonna have pi over to for the cause. We're splitting the pie into two pieces pi over two and one and then I'm gonna take this some as and goes from 1 to 10. Sorry. From 1 to 2. And what we have here is we've got the we're doing two different rectangles. So now we have to take the stuff. And what we're gonna do here is we're gonna take en Sorry, not tax end times pi divided by two look, and I'm gonna plug that into my calculator and that Give me the approximation for for two different rectangles. So pi over two times. Now the some as X goes from 1 to 2 of the sign of X. Hi. Over. We're gonna get an answer. That's approximately 1.5708 This is the approximate area under the curve using only two rectangles. Now we're gonna slightly change the formula in order to deal for five rectangles. The with, instead of being pi over two, will now be pi over five. Instead of having two rectangles, I'm now going to have five rectangles. And now again, I'm going to be splitting them up in into fifths for this some. So I'm going to go back to my calculator and I'm gonna change my summation notation in there to make sure that I have the right numbers. Basically, all of the choose will become five when will have approximately 1.933 eight as an approximate area under the curve to clean that up a little bit. 0.9338 We're gonna repeat that process one more time. We're gonna take the some from 1 to 10 of the sign of end pi over 10. And again, I'm gonna go back to my calculator and I'm going to replace all of my five with tens. And I'm going to get 1.9835 if I continue to add more and more rectum.

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