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Estimate the area between the graph of the functi…

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Problem 2 Medium Difficulty

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.
$$
f(x)=\frac{1}{x+1} ;[a, b]=[0,1]
$$

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Calculus 1 / AB

Calculus 2 / BC

Calculus Early Transcendentals

Chapter 5

INTEGRATION

Section 1

An Overview of the Area Problem

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Michael A.

May 9, 2020

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Video Transcript

So this problem We're working with the function f of X is equal to one over X plus one working on the interval from 0 to 1, we're gonna be looking at two rectangles, five rectangles and 10 rectangles. The the more rectangles we use, the closer area will be to my final. To my more precise answer, we can use the sun because we're gonna have Teoh take each of the different pieces to find each of the rectangles were Take the some his end goes from 1 to 2 of 1/1 over and notice I've replaced the X with one over And cause this would be the the With of the rectangles some will be the height of the of the rectangles. Sorry. The with here is gonna be one over And okay, so I'm gonna take the some multiplied by the withs and we will get our final answer. When I plug this into my calculator, I'm gonna get approximately 0.5833 This will be an approximate area under the curve. We using two rectangles when I use five rectangles. I'm gonna slightly change the formal from 1 to 5. I noticed. The only thing that changes is the number rectangles. The width of the rectangles will also change will be 1/5 of the of the value compared to half of the value. So once I take the some, I'm gonna be multiplying by this value here. So that's going to give me 0.71 which is much closer to the area under the curve, but not quite there yet. When I repeat this process for 10. Yeah, and make my with 1/10 of a unit. I'm gonna end up with 0.7980 I can continue to go. Aiken 6100. The more rectangles they used, the closer my area gets to the actual here.

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