Problem

Estimate the area between the graph of the functi…

03:13

Problem 4 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.
$$
f(x)=\cos x ;[a, b]=[0, \pi / 2]
$$

Answer

0.56,0.83,0.92

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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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Video Transcript

so in this problem were given a function F of X is equal to the coastline of X. Now, the co sign of X is working on the interval from A to B, where a is zero and B is pi over two. Now we're gonna split this up into multiple rectangles and the more rectangles I use, the closer my area approximation will become. So I'm gonna first simplicity's sake I'm gonna use to I'm gonna use five and I'm gonna use 10 to get an approximation. Clearly, the more rectangles they use, the better approximation might. My answer will become and the pat you should go to see the pattern from these first examples. Now the with right, we're gonna take pi over two and divide that by two. That's gonna be my with now to get the heights, all of the heights, there's only two of them. I'm gonna deal with some from 1 to 2 of the CO sign off. Now you have to be careful here. We've got high over two X over to, and when I enter that into my calculator, I'm going to get an approximate area of 1.3408 when I have five rectangles, all of the twos that air in the denominator here and here and here will end up becoming five. So I have pi over two, which is my and value have broken up into five pieces. Now I have five triangles and my co sign goes from Pi over two. And sorry the not next time using acts because expends up on your calculator. Um, pi over two n over five. When I approximate that value on my calculator, I'm going to get 1.1488 closer estimation to the actual area. Finally, I'm gonna take Pi over two and I'm gonna divide by 10. I'm going to get the summation from 1 to 10 co sign of pi over two n over 10. And when I plugged that in my calculator morning at 1.765 and this is

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