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Use the Midpoint Rule with the given value of $ n $ to approximate the integral. Round the answer to four decimal places.

$ \displaystyle \int^{\pi}_0 x \sin^2x\, dx $, $ n = 4 $

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00:35

Frank Lin

00:47

Amrita Bhasin

Calculus 1 / AB

Chapter 5

Integrals

Section 2

The Definite Integral

Integration

Elman R.

March 27, 2022

Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. ???? 2x sin2(x) dx, 0 n = 4

Missouri State University

Oregon State University

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.

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Use the Midpoint Rule with…

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Shown below.

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so everyone used the midpoint formula to approximate this. Um, what I first do is just figure out what Dell Texas. So over here we have the formula for that. So r B is pi r a zero and that our in is going to be four. Or at least let Chuck Yeah, in this for So there's just gonna give us pie fourth. So now there's two ways to do. Says we're doing the midpoint formula. Um, we could just kind of use like the equation that I have a top here or what I normally like to do is first create like the set of partitions, like we would be doing the left or right. Riemann sums and then just average them. So I'm going to do that. So it's going to start from zero and then remember to get the next X I. We just had Delta X two. It would be pie, fourth pie, half three, pie fourth and then pie. And so now, So this is just a set of partitioning, and then the ones for the mid point we just average thes, so the average of that would be pie eight to the next one will be three pie eight five Pi eight and then seven Pi aids. All right. And so now if we want So since Delta X is just gonna be pi for you just kind of pull that out front of this So it would be pie fourth times that we would just plug in pi a three pilots and so on. So high eights times sine squared of high plus three I ate signed square of three pie eight and then plus five pi eights plus plus sine squared of five high eighth and then plus seven pi eight sine squared of seven pi tight. Right Then we just need to plug this song into a calculator and then get our estimate. So let me see what we get. So it's like approximately so pi fourth times, uh, 3.141 592 And then just gotta keep on going that we would need to multiply this by pie and then divide by four. And then this would be approximately 2.467 Or and I believe they wanted it to go to four decimal places. So this will be approximately integral from zero to pi of x times sine square of X x

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