💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Like

Report

Numerade Educator

Like

Report

Problem 55 Easy Difficulty

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

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

Answer

$\int_{0}^{4}(x-2)^{2} d x \geq 0$

Discussion

You must be signed in to discuss.

Video Transcript

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