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(a) determine the average value of the given function over the given interval, and (b) find the $x$ -coordinate at which the function assumes its average value.$$f(x)=x e^{-x^{2}},[0,2]$$

(a) $\frac{1-e^{-4}}{4}$(b) 0.262997 or 1.287403

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 7

Substitution and Properties of Definite Integrals

Integrals

Missouri State University

Campbell University

Harvey Mudd College

Idaho State University

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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(a) determine the average …

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Find the average value of …

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Determine the average valu…

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Okay, So an average value problem, What you want is to go from the bounds that they give you from 0 to 2. But in front, you want to do one over subtracting those two values and then you can plug in the equation X e to the negative, X squared DX. Uh, and then from here, you can use u substitution. U equals negative X squared. So do you will equal negative two x dx. And as you look at my work, what I technically do is have my students divide that two X over. So then what I can do is simplify this problem. To be so one half, because to minus zero is too. Um, I'm gonna change my bounds, though, because I need to change in terms of you. So negative of zero square two so zero to square to before so negative four there. And then what I can do is I can I can rewrite this as e to the u power. But what I'm gonna have is negative one half d you to replace this DX. Uh, and these Xs are canceling each other out. Hopefully that all makes sense. Otherwise, you might want to do a little bit more. The other thing I would do this is just me is I would pull that one half in front one half times. One half is 1/4. And then I would also flip my bounds around because of the negative. You can do that when you have a negative. So flipping these cancels out the negative. I guess I should raise it that way. So our anti directive is 1/4 e to the U power from negative 4 to 0. So we're looking at E to the zero power, which is one minus, he to the negative fourth power. And instead of multiplying by 1/4 I'm going to divide by four. So this is your first answer to part pay. Now, the second part, you're going to need a graphing calculator for because this is a numerical value. This is just a number. Some number. So what I would do is go to a calculator and type into why one or if you're using, does knows, you might just type in y equals X E to the negative x squared. And then why to that number now, I would actually type in like this and you're going to look for your point of intersection. And what you'll find is the point of intersection. Is that an X value? There's actually two of them. 0.263 Uh, and we only care about the interval from 0 to 2. Okay, we don't care about outside, so if it crosses outside of that interval, we just ignore it. 1.287 And I liked around to three decimal places as well. I stopped right there. So here's your answer to Part B. Hey, and be

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