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

$0 \cdot 549306$, 0.453946, 0.420535

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 looking at the function F of X is equal to the natural log of X. Okay, we're looking at the interval in this case from 1 to 2. Now, the total distance from 1 to 2 is one unit. So I'm gonna take that one. Unit one, take that one unit we're gonna be working with rep two rectangles, five rectangles and 10 rectangles. The more rectangles I use, the more precise my answer will become in this particular case, what I'm going to do here is I'm going to take the with times the sum of all of the heights that will give me the approximate value. So my original with his one, and I'm gonna divide that in half. So each with will be one half the unit. If I was doing five rectangles, each with would be 1/5 of a unit. And if I did 10. If you 1/10 of a unit. Since I'm doing two rectangles, I'm gonna have the sum of the two different rectangles. And I'm going to take the natural log of one plus because again, I'm starting at one all of the X over twos meaning one half and two halves. So when I plug this into my calculator, I have one half times the sum from one to chew of the natural log, Uh, one plus and over to star. That's it, then. Not an ex apologize, but an over to and that will give me an approximate area of 0.5 for 93 When I repeat this process, I'm gonna just switch all the twos to become five. So in this case, I am going from 1 to 5. Meaning I have five rectangles. It's the natural log of one plus all of the fits. 1/5 2 fists, refits, forfeits and five fits. This will give me an approximate value of as I plug it into my calculator. 0.4539 Repeat this process one more time and again, I can continue to do this process as many times as I want with the more and more rectangles that I use, the more precise my answer will become. So in this case, I'm gonna have the some from one to 10 meaning 10 rectangles to get him starting at one, and I'm gonna add all the texts This will give me an approximate value. 0.4205 The more rectangles I do, the closer I get to my actual.

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