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Determine the area of the region between the given curves.$$f(x)=x^{3} \text { and } y=8 x$$

32

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 8

Applications of the Definite Integral

Integrals

Campbell 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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Determine the area of the …

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Find the area of the regio…

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uh What I would do to do this problem is a first of all thing that X cubed is a function is an odd function that goes on both sides. Um So why it was execute, but then we also have this function Y equals eight x. And so there's two points of intersection, why equals eight x. Um And so the two areas are actually going to be identical to each other. So what I would actually do Is find the area of one of those regions from X equals 0 to another piece. I haven't figured that out yet and then just double that area because it's going to be the same on this side of that answer. So we can figure out where that point of intersection is by setting those equations equal to each other. And we can solve that by subtracting the eight X. Over and we can factor out an X. So we already saw the X equals zero but the other one b uh X squared it was eight as plus or minus the square root of eight but just like I mentioned, you could have both the positive and the negative nature. Um So I'm just gonna double that answer. So I mean take the upper function which is that eight X. Function and subtract off the lower function which is the XQ function. So then the integral of that. So then double would be adding one to the exponent divided by the new expanded eight by by two is 4. It's 1/4 x to the 4th. D. X. R. C. from 0 to Room eight. So it's really nice about this guy. I think this is nice Is when you square root eight squared just cancels out the square root. So when I plug that in it's just gonna be eight times four was just 32. And the same thing happens here except it's uh route eight to the fourth. Power will be the same thing as eight squared. uh what should give me 64, divided by four, would give me 16 and plugging in zero, And for both of those, you really should still do it, but it's going to be 0 0 so it's not gonna change anything. So as you're looking at, your final answer is going to be double 16, Which gives me 32. Yes. Mhm.

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