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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 $
$\int_{0}^{4}(x-2)^{2} d x \geq 0$
01:29
Frank L.
00:19
Amrita B.
Calculus 1 / AB
Chapter 5
Integrals
Section 2
The Definite Integral
Integration
Campbell University
Oregon State University
Idaho State University
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Evaluate the integrals.
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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