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Use the properties of integrals to verify the inequality without evaluating the integrals.

$ \displaystyle \int^4_0 (x^2 - 4x + 4) \,dx \ge 0 $

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01:29

Frank Lin

00:19

Amrita Bhasin

Calculus 1 / AB

Chapter 5

Integrals

Section 2

The Definite Integral

Integration

Missouri State University

Oregon State University

Harvey Mudd College

Baylor University

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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Use the properties of int…

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Use the properties of inte…

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Verify the inequality with…

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All right, let's move on to this question. We are given the integral off X squared minus four X plus four D X, and then we are required to prove that this is a positive number. Or maybe go to zero gay using the properties that we've learned about integral. Well, this is an integral in an integral represents in area under the curve. So in this particular case, what's so special about ffx? It's the fact that you can factor this nicely as X minus two quantity squared. And then what we know about this guy is that this is a perfect squared. So no matter what value of X we pick, it's going to be non negative. So this is greater than or equal to zero. So because we're finding the area under the curve off a number that it's never negative, we know it's integral is also going to be non negative. And then this is how you prove this problem

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