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Evaluate the given definite integral.(a) $\int_{-1}^{2} 2 x^{2} d x$

(a) 6(b) -6

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 6

The Definite Integral

Integrals

Campbell University

Idaho State University

Boston College

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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Evaluate the definite inte…

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Okay, so this is basically the same problem written in two different ways from negative one 22 of two X squared DX. And then the second one. Yeah, I label. They just switched the balance two x squared debts. So the nice thing about this is the anti derivative is the same. No matter what, where you add one to your exponents. Two plus three is one, and you divide by your new exponents, and this one goes from negative 1 to 2, whereas this anti derivative again is the same. But it's going from two to negative one. So the difference, Um, and let me just factor out that two thirds, you know, you can do this in with the anti derivative, because technically, what you're doing is just taking two and cubing it. Um, two times, two times two is eight and then minus plugging in negative one in Cuba in which would be negative one. Um, well, the only thing that's different over here is you're plugging negative one in first. So negative one cube is negative. One minus two. Cubed is eight. Um, so I hope you are seeing the difference here. And the only difference is that you switch the bounds around. Um, so the final answer for letter A as you get nine times two thirds. So nine times two is 18. Divided by three is six. Whereas over here you have negative nine times. Two thirds. Well, negative. Nine times two is negative. 18 divided by three is negative. Six. So the moral of this story. So this is a letter A And this is the letter B, um, is that if you switch the bounds around, you get the same answer, but negated once positive and one negative.

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