Problem

Estimate the area between the graph of the functi…

02:48

Problem 8 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.
$$
f(x)=\sqrt{1-x^{2}} ;[a, b]=[-1,1]
$$

Answer

1,1.42,1.52

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Calculus 1 / AB

Calculus 2 / BC

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Chapter 5

INTEGRATION

Section 1

An Overview of the Area Problem

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Video Transcript

So in this problem, we're looking at the function F of X is equal to the square of one minus X squared, and we're looking at the interval from negative 1 to 1. Now, the total distance of the interval from negative one to positive one is two units. Okay, so what we're gonna be doing is we'll be looking at two rectangles, five rectangles and 10 rectangles. Now we can use more rectangles, and the more rectangles I use more precise my aunts will be. But I'm just gonna get you the pattern here, and then you can use your calculator to find more precise answers. So in this particular case, the first thing that we want to do is you want to take this distance of two and find out what the widths would be because we're gonna take the width of the rectangles, which will be constant and multiply it by the output values or the heights of each of the some of the heights of all of the rectangles. In this case, I'm gonna take two and divide it into two equal pieces. I'm gonna take two and divide into five equal pieces and take two and divide into 10 equal pieces. This will be the width of each of the rectangles. Now we just have to have the sum of all of the heights, and then that will give us the approximation for the area under the curve. In this particular case, we're gonna have the song as an goes from 1 to 2, meaning I have two rectangles and then we have the square root of one minus. Now, here's where things get a little challenging. I have negative one is my starting value, right. And in this case, I'm going to add two X over twos. And what that's going to do is this is going to be calculating where the outputs will will be, which X values will give me the outputs. Now, when I when I approximate that value, is gonna proximate toe about one. Not a great approximation for this area under the curve, But the more rectangles I used, a more precise my answer will be for five rectangles. I'm going to go from 1 to 5. That's five different rectangles and then or take the square of one minus and again, negative one is my starting value and in this case, I'm going to add all the two fits 2/5. Um, forfeits, etcetera, etcetera Center. Because we want to get to the total of two. This will give us approximately 1.438 4238 This is the approximation. With five rectangles, we continue the process with 10 rectangles. One minus. Sorry. These air squared one minus negative. One plus two x over tens. We're gonna get all the two X over tens for all the 10 rectangles squared. And this will give me an approximation of 1.5 one aids five. And so the more rectangles I use if I change this 200 this would change to 100. This would change 200. This would change 200 I would get more and more precise.

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