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Numerade Educator



Problem 24 Medium Difficulty

Evaluate the integral.

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




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Video Transcript

Let's evaluate the integral from zero to one of the square root of X minus X square. So looking inside the radical we see x minus X squared. And this is not of one of these forms that we'd like to use the tricks of right away a square, plus or minus X squared or x squared minus a square. So what we should do is go ahead and complete the square. So taking our expression inside the radical, maybe pull out a negative sign, then I could go ahead. And so we have Ah, negative one in front of the ex. Then we divided by two and then we square it and that gives us one over four. So we add that inside. And really because of this minus, I know here we really secure subtracting negative one over for so we have to make up for it by adding it back. So we completed the square and we could write. This is it's minus a half square plus one fourth. Or if you like, we could get one over for as one over into squares minus x minus the half square. Now, if we take a use of U equals X minus one half This becomes what happened Slur minus you squared And this is what we would like if we're going to do it. Tricks up. So before we do that, let's go ahead and express this integral in terms of you So we have the integral we'LL have to find the new limits of integration And then the square root of one half squared minus you square again This is because we we simplified the expression inside the radical X minus X Where is equal to one half squared minus you squared says all we did was replace the expression inside the radical. Now, because this is a definite no girl we'LL have to find the new limits So X equals zero Plugging that in here into the use up you get u equals negative a half and then plugging in the upper limit one you have u equals one minus the half which is the half also to you is equal to DXE Halt. So I'm running out of room here so I'll go to the next page. Our previous integral was this. Now we're in the position to do a trick So here we should think you'd be one half sign, Daito. So here is one half. Then we have to. You is one have coast and data. And once again, because we have a definite integral, we should go ahead and find the new limits. So observe that when you have a tricks up of the form a sign data, we should always take theater to be between negative, however, sue in private. So plugging in you equals negative a half We're negative. One half equals one have signed data, which means signed data equals negative one. And the only time that happens in our interval is that negative numbers who and now for the upper limit one half plugging that in for you one half equals one half signed data. So sign data equals one. And the only time that happens in this interval is that Piper too. So these will be our new limits of integration in terms of data. So before we go on, we should just simplify this radical. So this is the square root of one over four, minus one over for science square fat throughout the one over four. We're left over with one minus sign square. squared of one over for is one half. Then we have square root of co signs where which is one half cho santa. So let me go to the next page. We have a rule negative five over two pi over too. And then the radical became when have co signed data and then d X was also one have consigned data. So remember this was DX. Let's go in and simplify this. Let's pull the one fourth out of the radical and then we have co science where? Let's go ahead and use a half angle formula for a co sign squared. So this is one plus coastline to data all over too. So we have won over eight, to which God that comes from the two times for and then we have one plus coastline to data data which we could integrate in. Uncle of One is just data and in a rules coast on to data assigned to theatre over too. And these are end points. So plug these in dated equals pi halfs. Then we have signed a pie. And then when we plug in negative, however, sue and then we use the fact that sign of pie inside of negative pyre. Zero We have Pirate Sue plus by over two. So that's just pie. And then we have one over eight. So our answer is I divided by