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Problem 51 Easy Difficulty

Prove Formulas 9.

Answer

$\bar{y}=\frac{M_{x}}{m}=\frac{M_{x}}{\rho A}=\frac{1}{A} \int_{a}^{b} \frac{1}{2}\left[f(x)^{2}-g(x)^{2}\right] d x$

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Video Transcript

So in this problem, we're going to be driving this Formula nine and then we derive formulas. It's helpful to take the more complex side of want to prove and transform it into this simpler side. So in this case will be working with the right hand side to simplify it into the left hand side. And to do this, we're going to be using Formula One from this section to help us take steps to supply. Because we have a product of two signs were able to use Formula One and it becomes 1/2 times co science Alfa plus beta over to minus Alfa minus fada over to minus co sign Alba Spada over two, plus Alfa minus beta over to Yeah, we got a lot of fantasy is here. All right, so now we can begin simplifying this further. So the negative to and the 1/2 can simplify to create just naked of one, so we'll put a negative sign out here. Then we have a co sign, but we can also simplify what's inside of the coastline the office attract. We have Alfa minus Alfa, and we have beta minus negative beta. So that gives us two beta divided by two and then we have minus again. Added together. What's inside the co sign our Betas Cancel and we're left with two alphas on top and the two on the bottom. And now let's simplify each of these. So we have co sign of beta minus co sign outset and then distributing through the negative co sign, Alfa becomes positive. So we have course Alfa here and then we have minus co signed beta. This is equal to the left hand side. So now we have officially verified this formula.