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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.$$f(x)=\sin ^{-1} x ;[a, b]=[0,1]$$

0.549306, 0.453946, 0.6578

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 we're looking for the area under the curve and we're looking at the function in verse Sign of X on the interval from 0 to 1. Well, if I started zero and I go toe one that tells me that the total distance traveled is one I'm gonna be splitting these up into two rectangles, five rectangles and 10 rectangles. I could keep going to 6100 but more rectangles I used, the more precise my area under the curve will be, I need in order, find the area under the curve. I'm gonna multiply the with which, in this case, if I do two rectangles, I take my total distance, which is one and divide it into two units. Take my total distance, which is one divide into five units and take my total distance one and divide into 10 units. I'm gonna take that with and I'm going to multiply by the output values at each of those intervals. So I'm gonna take the inverse. Sorry. Back up a second. I'm gonna take the sum of all of the output. So from 1 to 2, because there's only two rectangles from 1 to 5. Because There's five rectangles from 1 to 10 with because there's 10 rectangles. If there were 100 would you 1 to 100 etcetera, etcetera, etcetera. We take the inverse sign of all of the haves. One half and two have so be the output value that one half and the two abs in this case, we're gonna get an approximate value of 1.0 for seven to when I plug in five rectangles, I'm gonna get the inverse sign of all of the 5th 1 52 53 54 55 fits. And this will give me an approximate value of 0.75 09 The more rectangles they use, the more precise my answer will become that I can do the inverse sign of X over 10. All the 10th 1 10 to 10 3 tents, etcetera, etcetera, etcetera. In this case, I'm gonna get an approximate area of 0.6578 The more rectangles that I use, we can continue this pattern for more

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