Problem 11 Medium Difficulty

Evaluate the integral.

$ \displaystyle \int_0^{\frac{1}{2}} x \sqrt{1 - 4x^2}\ dx $

Answer

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Video Transcript

Let's evaluate the integral from zero to one half of X times the square root of one minus for X squared. You can actually do this problem using the U substitution, just like you would be the thing inside the square root. Let's not do that method here because this is seven point three. This's the trick substitution section. So let's do insurance substitution here. So, looking at the inside of the radical, we see that we have something that's very similar to a square minus X where except here. We just have a number in front of the X. That's not a big deal for us. If this was the substitution, you would normally take X to be a sign data sonar problem. We have one squared. So is one minus two eggs, all in parentheses square. So we should take the thing. The variable being squared, which is to X and said that equal to a which is one signed data. So that's their trick. Then we could differentiate first, I guess here you could just divide by two, and then we could differentiate T X is coastline data over to and because we're dealing with a definite enrol. We can go ahead and try to switch those limits in terms of data. But if we're going to do this, we have to make an observation here. And is that when you do a tricks up of this form when you sign, you have to put a restriction on data. And the restriction is that it's between negative power over too, and powerful too. This is actually gonna help us because they will let us find the new data values. So the original value lower limit was X equals zero. So now plug in X equals zero into this equation up here and saw for data. So we have zero equals scientific data over to this means sign a zero and the only time sign a zero in this interval appear is one date a zero. So this is our new lower limit. Similarly, we could find the new upper limit corresponding to one half. So let's behalf, then, plugging this into this circled red equation. Up here we have one half equal sign over too. This means sine equals one. And the only time that sign is one in this interval appear. Is that pie over too so data equals. Perhaps that's our new upper limit. So we can try to write this new integral in terms of NATO. So we have the General Zero pirate too. Next watch from over here We know that society data over to and let's go to the side and simplify this radical Here you have square room one minus four X squared In our case, let me come down here. This will be one minus four Science Where? Data over four. So we get Cancel those force We have one minus science Where which is co sign squared and a squirt of this is closing. So that's the radical. So we have times it has come down here co signed data and then we have DX which was co signed data over to later. So right now this is our new instagram. So it's separate this from our scratch work. We'Ll simplify this a little bit. I could pull out the four outside the integral The one fourth zero for two Now we have close eyes. Where? Time sign now since I'm running out of room Let's go. Let's pick this up on the next page we have one fourth in a girls, They're up to coastlines. Where? Time sign. Now this interval. We should go ahead and do a U substitution. Let's knew the coastline. Then negative. You will be scientific data and once again before with the definite local. So we should switch those limits. So data equals zero that corresponds to you. Nipples coastline zero people's one upper limit was fired too. So data equals Hi, Sue corresponds to U equals co sign too. And we know that zero. So after are you so we can write this as a negative one over four. The negatives coming from new substitution in enroll from one zero useful here to you. And if you don't like the fact that the lower limit is bigger than the upper limit, you could go on and switch the limits. And when you do so, that introduces a negative sign, and we could cancel with the negative sign that's already here. So you have one fourth in a girl's ear. The one you square, do you? And now we're ready to evaluate. Yeah, you're cute over as well. From one zero. Plug in those end points and we have won over twelve. And that's your final answer.