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(a) determine a definite integral that will determine the area of the region shown and (b) evaluate this integral.

(a) $\int_{2}^{3}\left(x^{2}+1\right) d x$(b) $22 / 3$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 6

The Definite Integral

Integrals

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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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(a) determine a definite i…

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(a) Set up an integral for…

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Construct and evaluate def…

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So, given the graph in this problem, when we're looking at X squared plus one and so that's what they give you. And they also give you these other values like X equals. I'm not paying close attention to where I'm putting this, like maybe excess ones right there and access to is there an excess three is right there. But we're looking for the area under the curve. And what you do is you consider your bounds. So mathematically it's the integral of X equals two X equals three of the function that they gave you, which is that X squared plus one DX. So that's your first answer to part egg Just setting up this equation. And, uh, I guess I can just kind of point out that exit goes to Mexico's three. Those are your bounds is the vertical lines and then the function it goes right there. So now we're ready to find the anti derivative. This is part B, by the way, where you add one to your exponents and divide by your new exponent. You can also consider what would be the what? Uh huh. I guess I should say that the derivative of one third X cube is X squared, and the derivative of X is one we're considering from 2 to 3. Um So as I looked at this problem, I start plugging in my bounds. Well, instead of doing three times three times three. As I plug this in, I can think of one of those threes. Cancel. And three times three will be nine plugging three here three and then subtract off. Plugging into two cubed is eight thirds plus two. Now, if I were trying to do this in my head, I would actually, um, distribute this minus in here. So I'm looking at nine plus three is 12 minus two is 10. Um, but I want to write down 10 is 33rd because I want to get the same denominator as eight thirds. And that way I can do 30 minus eight. My head to get 22 3rd should be your correct answer for part beat. And it is this is your answer to be. And then up here is your answer to a

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