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Determine the area of the region between the given curves.$$f(x)=2 x^{2}-3 x+4 \text { and } g(x)=-x+8$$

9

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 8

Applications of the Definite Integral

Integrals

Campbell University

Oregon State University

University of Michigan - Ann Arbor

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 between the …

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Find the area of the regio…

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all I know in this problem is that we have a parade below. That's opening up. Don't really care. It's more on the right side of the left side. But then we also have this other function. That is a linear function. So this would be G This pdf. What They don't tell us the problem is where these two graphs intersect, but we can figure that out by setting them equal to each other. So that would be two, X squared -3. x plus four would have to be eaten. That's F by the way, is equal to negative explicit. That's cheeky. And what I can do then, Because it's a quadratic, I can add one X over here. I can subtract eight till left side. So the negative forward, I can set it equal to zero and I can factor out at two. So it's like I'm dividing everything back to -2 is equal to zero, and then I can find factors -2 Plus one. So that would tell me, I don't know if I really need to show all this work that my points of intersection Would be at x equals two right here and X equals negative one. But all I really care about is that now we can do the inter cool From -1- two. And it's actually gonna be the same as this graph except the upper function as a linear one. Uh So I should have if I wanted to make this perfect had negative X. And then subtracted these three terms to the right side. Uh And all it's going to happen is it's going to be engaged everything I had earlier. So the negative two X squared plus two X plus four dx. Now I'm ready to do the anti derivative. We should be adding one to the exponents. Multiply by the reciprocal of your new exponents. So choose to cancel their From -1, 2. I can plug in my upper bound now. So two cubed is eight times that native to his name. 16/3. Two squareness four. four times 2 is eight. And now I need to subtract off plugging in a lower bound which is negative one, negative one cube is still negative. So that's going to change the sign negative one squared is positive one and four times anyone is negative four. Now I would first simplify these before because I'm just doing the math in my head that four plus eight is 12 -2/3 and then that would be negative three of if I distribute that become plus three. So I'm looking at if I combine this piece with this piece, I'm looking at negative 18 3rd which is equal to negative six plus 15. 12 plus three. Just adding any terms. I get an area of nine which is the correct answer.

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