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Using Riemann sums on a spread sheet, the midpoint of each sub-interval, and a large value for $n,$ find an approximate value of the integral estimated in Example 6.

$0.32174914(\text { using } n=100)$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 6

The Definite Integral

Integrals

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

02:07

Calculate the left Riemann…

03:04

Compute the left and right…

So for this problem, we want to compute our left and right remains, um, for our function fx on our interval from 0 to 6. And so our first up in computing any ransom is going to be computing our Delta X. And so the way that we do that is we are going to subtract our final end point minus our initial in point. So that'll be six minus zero over interval, and we divide this by the number of sub intervals. So in this case, since we're finding of the left and right room right hand Riemann sums with 67 levels each. We can just use the same delta X of one Do you can clear each of these remains Selves. And so next we're going to be able to choose all of the X coordinates that we you want to use for this problem. So essentially, we're just going to start at the beginning of our interval, which is in this case, zero. And we're going to incriminate by one all the way up until the other end point. So that means that our X coordinates will just be a 0123456 Our next up is going to be computing affects for each of these terms. So, for instance, if we look in F zero, we get three minus three months zero which will just be three months. Three, which is zero. I'm so if we can keep this for our remaining terms, we're going to get a 12321 and zero. And from here, we have all the information we need to compute. Both are left and right, or even sums. So we're going to start by computing. Our left hand remained so And the way that we're going to do this is we're going to look fire Delta X. So in this case, it was one by the sum of our 1st 6 y coordinates, or FX values. So that means that we're going to add zero plus one plus two plus three plus two plus one. So this is going to end up being equal to nine, and we can have the same thing essentially for our right hand reading some again. We're going to multiply Delta X, but this time we're going to multiply it by the sum of the right six terms. So instead of starting at zero, we're going to start at one. So we're adding one plus two plus three plus two plus one plus zero and again we get an area of nine. And so the next seven, this problem is going to be comparing this to the actual area under a curve so it don't make a sketch of our graph. We know that it's going to be an absolute value growth with its vortex at the 0.33 And it has eggs intercepts at zero and six. And so we know that to compute this, we're just going to find the area of a triangle, which is 1/2 times our base, which is a leg of six. We said times our height, which is three, so we can see that our area is equal to have 18 which is just nine. So our left hand remains, um, our right hand rheumatism and our actual area under a curve are all equal to you nine in this case,

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