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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.$$f(x)=\frac{1}{x} ;[a, b]=[1,2]$$

0.58,0.65,0.67

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 this problem we're looking at is we have a function F of X is equal to one over X were working on the interval from one to two. Now, I want you understand that in order to, uh and we're gonna be working with two rectangles, five rectangles and 10 rectangles and the pattern could be continued for as many rectangles as you would like. The more rectangles you use, the closer your approximation will become. Now you have to understand that when I split up the distances from 1 to 2 by twos, for example, here's one here's to and I split into two pieces. Right. It goes one on 1.5 to if I split into five pieces, right, it's gonna be one one in 1/5 one in 2/5 one in 3/5 one in 4/5 and two. And so again, we want to add up these values, and then we want to multiply by the with which in this case would be one fit. Right. So what we're doing is we want to evaluate the output at these values times the width, and that will give us the some of the rectangle. So in order to do that, what I'm going to do here is the with in this case, for when you have to write from 1 to 2. If I split that one unit in half by with is one half and then I'm going to add up all of the outputs, right? So in this case, I'm looking at the some as X goes from 1 to 2 because only having two triangles right and what I'm going to do here is I'm going to again. We're starting at one. So in this case, we're gonna have one over right one plus x over to cause we're doing the haves one half and two halves. And when I evaluate this in my calculator, you'll see that the area comes out to be approximately 0.5833 Now, when we repeat that process, what we want to do here is in this case, when we break it up into fifths, we can see that the with will be fifths. We want to add up this all of the rectangles, the output of all the rectangles which will give me the heights. And in this case again, we're starting at one. But we're adding 5th 1/5 2 fifths, 3/5. And when I evaluate that we're gonna end up with 0.6 points is 0.6456 and we're getting closer and closer to the actual value. When we have 10 rectangles, the width of the rectangles will be 1/10. We'll have 10 rectangle. That's what this means. And again we're gonna be adding the output when at each of the tents. So x over 10 we re evaluate this guy, we're gonna have 0.668 eight. And again, what we can do is we can replace the tens with 100 with 1000 with whatever you want and the more rectangles I get, the more precise my area will be.

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