Problem

Left and right Riemann sums Complete the followin…

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Problem 28 Easy Difficulty

Left and right Riemann sums Complete the following steps for the given function, interval, and value of $n.$
a. Sketch the graph of the function on the given interval.
b. Calculate $\Delta x$ and the grid points $x_{0}, x_{1}, \ldots, x_{n^{*}}.$
c. Illustrate the left and right Riemann sums. Then determine which Riemann sum underestimates and which sum overestimates the area under the curve.
d. Calculate the left and right Riemann sums.
$$f(x)=\sin ^{-1} \frac{x}{3} \text { on }[0,3] ; n=6$$

Answer

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Calculus 1 / AB

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 5

Integration

Section 1

Approximating Areas under Curves

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Video Transcript

Okay, so I just did a quick Google search saying Dez, most Riemann sums and the very first thing that popped up was a program that you could just type in the equation. So I went ahead and did that already. Inverse sign of X over three. And give your left endpoint, which already typed in equals three. You're right in point vehicles so equals zero B equals three. And then you could tell how many rectangles you want in this picture. So I choosing six. Now, you might be looking at this and saying, You don't really see the rectangles. It's this First one is really small. The second one stops it. X equals one and 1 to 1.5. And this shouldn't make sense that your change in X is 0.5. Because if we're going from 0 to 3, you subtract your bounds and you divide by six. Then each width needs to be 0.5. Now, this next thing you either type in zero for left handsome one for right, handsome or 10.5 for midpoint. I'm just gonna go ahead and do zero first because they want the left hand. Some is your answer first. So as this neat little formula for you will tell you what you're correct, answer is. And what this does is it saves you from figuring out what in verse sign of zero is which is actually zero, which makes perfect sense. Um, saves you from typing an inverse sign of 0.5 divided by three. Um, you know, because this value 0.167 you'd have to feel like it's not on the unit circle. This one is. I could have figured out what inverse sign of 1.5 s. Mhm. Okay, um, these values are what you do. So if you're doing this by hand, you would have 0.5 times zero plus 0.5 times 0.167 That's the area of this rectangle plus 0.5 the area the width of this directing all times 0.34 plus 0.5 times per five or six. That's the area of this rectangle. It's really it's not outlined. Plus 0.5 times 2.73 plus 0.5 times 0.95. So if you have a teacher that needs to see that work, that's what you would do and your answer is 1.373 So let's do the right remind some, which I just have to change. That could be one. And all of those values that I was talking about earlier are the same. But the difference is you don't do 0.5 times zero now because I was the left hand you 2.5 times pi over two and add everything else together and notice how my son changes to 2.158 It's a larger value, so thanks for watching.

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