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Sketch the area represented by the given definite integral.$$\int_{0}^{2} 3 x^{2} d x+\int_{2}^{8}(-2 x+16) d x$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 6

The Definite Integral

Integrals

Campbell University

Harvey Mudd College

Baylor 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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All right, so we're asked to find the graph of this problem. And, uh, basically, what the scrap is going to look like is a piece of ice, because there are two pieces of peace right here and then the other piece from 2 to 8 of negative two X plus 16. So if you're familiar with piecewise functions, this problem would probably be pretty easy, because these values the bounds are calling bounds because their boundaries. So if you look at the graph, which I'm going to just make my x axis over here uh, sorry. Why access? I said that wrong. But if we were to look at 12 and you can see I have to go all the way to 8345678 Um, what you can do is actually just plug in those numbers into that function and see what answer you would get. So zero square 20 times three would be zero still, um, Now I'm not going to label my X axis because I tend to do a really bad job of doing that. But if I plug in to and for X two, squared is four times three would be 12. So this order pair should be to 12. And I think all calculus students know the behavior of a parabola. Uh, if you're not quite sure, plug in one, and you would see that order pair would be 13 So we have an upwards u shape. I guess I should also label 00 over here. Um, and then we stop. So that finds the area under the curve from zero X equals 02 X equals two. So let me switch over to this graph. Well, we don't start drawing in until X equals to this bound. If I plug into and for that X, you would see that ordered pair would be negative four plus 16, which is the same point. And if I I hope you realize that this is a linear function link down to write one. But it might make more sense to plug an eight in here. Uh, native, two times eight. Snake of 16 plus 16. We give me zero. So 80 would be a point on the graph. I'll switch back to green. Um, and it's a linear function as a straight line down to write one, But again, I don't label my Y axis, so it's not very clear. If I did, I would have to try to make it so. 123456789 10, 11 12 Trying to match that up and again. We're finding the area under the curve from 2 to 8 on that piece, and this is the shape of that curve. So again, it looks like a piece wise when it's all said and done, and that's your answer.

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