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Evaluate the integral.

$ \displaystyle \int_0^{\frac{2}{3}} \sqrt{4 - 9x^2}\ dx $

$\frac{\pi}{3}$

Integration Techniques

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Let's evaluate the integral from zero to over three of the square root of four minus nine X squared. Now let's look at this instagram. This would be a little easier to work with if it was of the Form four minus Explorer, because this is two squared minus X where, and we would like to be, in the case, a squared minus X squared. But we're not in this case because there's a nine in the front of the ex. So what we can do is rewrite this four minus nine X squared is four minus three eggs square. Then we could do a U substitution. U equals three eggs. Do you equals three d. X so that do you over three and the limits of integration will change because we did a use up, so X equals zero was original lower limit that's going to correspond to U equals three times zero zero, and the upper limit was ex peoples two or three. And that's going to correspond to the upper limit. You equals three times to over three, which is too. So this interval can be written as wonder. The third's coming from the use of integral we have our new limits. Sierra two square room four minus three X squared, which is use cleared. Do you? So now we have this new integral and this we have a four minus you square. So this suggests that we should use the substitution you equals to sign data. Then we have Do you equals to co sign Data Diferente. And we have two options to ways to proceed here because this is a definite integral one way to proceed which won't will use next, is to find the new limits of integration through the truth Substitution. If you do go this road, you won't have to draw the right triangle because you'LL never need to go back to the variable you after the train. So so here. Let's observe that when you make a tricks up of this form that you're putting a restriction on data. So this is from the textbook. We have data's between negative power over too, and pirates who so the lower limit before was external. Excuse me, You, after we did use of was you equal zero that's going to correspond to plugging this into the tricks of equation. We have zero equals to sign data. So this means sign a zero and the only time sign a zero in this interval. Negative pion attitudes of poverty is one date a zero. So this will be our new lower limit of integration. Not so the upper limit was too. So you equals two. Plugging that into our tricks of equation again, we have to equal to send data. So scientific is one and the only time sign is one in this interval is one date aspirants also let's rewrite our instagram. So here we have square root of four minus use where this is four minus four science where we can fact around a four and then pull the four outside the square room that becomes a too. We have one minus science where? So we have a two square room Cho Science fair and we know that the square root of co sign squared it's just co sign. So we have to co sign there. So let's plug everything in when one third integral zero department to our new limits. We've just evaluated the four minus You squared in the radical and we had soup coastline data, so there's a too close and data there and then do you? It was also to cause and data. So we have. So now let's rewrite this. You have four over three in a row. Zero parable too, of course. Signed Squared of theta. And since I'm running out of room, let's stop right here and then go to the next stage. So let's rewrite the previous expression we have for me in a girl. Zero, five, two coastlines Where now recall for this trick in a metric integral. And the identity that we should use here is ko sense where they are equals one plus cause I'm too later, all divided by two. So if we use this identity here, we can cancel off this too with the four. So we get a two over three in a girl, one plus coastline to data. And now we could evaluate General. And one is just data and the integral of coastline to data the sign teeth over too. My help you to use a but this is our obvious. The underworld coast to data. You gotta use use up there to evaluate that. Now we could go ahead and plug in the end points so It's plugging power to first sign of two times pi to sign a pie. And then when we plug in zero, we give zero wass sign zero over, too. Now we know that sign of zero zero. There's zero. We couldn't go on that sign. If pie is also zero. So we're left with to over three pi over two multiplied in that simplifies just fire on three. So there's our answer.