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Find the exact length of the curve.
$ y = \sqrt{x - x^2} + \sin^{-1} (\sqrt{x}) $
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According to the ocean, we need to find the ark land we have the given question is R is equal to tu minus so scientific leg we after all you don is a new function off tu minus those scientists We know that the using different vision video is back Totota Then we differentiate dysfunction there is up Daschle Puta We know that the differentiation off to with respect to theta is zero their differentiation of scientific physical to cause data This is a minus two cause tutor when would be fresher dead afford Therefore we have the Arland The our land is as if he could do zero Dubai route half dash of the whole square and plus app off $2 square We know that when we applied this we have the value off app off death row Polska When we hold scared distance we have four course square teeter We have four cause square to down Bless tu minus Does sign data whole square digital When we sold this town we have as is equal to zero to buy When we sold this time there is four close square Peter plus four minus air Sign Duda Bless four signs square Buta Digital When the soul and simplifying the question we have the value off s has become zero to Dubai route it one minus sine theta Did you don't reuse, Integrate? Calculated Oh, Then we use integrate calculator the simplifying the value as we have as is equal do 16.
According to the question, We have to find our cleanse. We have the given from Chinese Apple Duda is equal toe sign Trita We know that we differentiate dysfunction with respect to Duda with respect potato Then we differentiate this function with respect to theta We have have Daschle Dita you know that the definition off scientist is called strata We have two equals three cheetah on Then we differentiate this angle This angle is treater There is three the wrist Rezko's trita therefore dark lend do our land He's as his equal do zero to Dubai limit Rude after dashed it The whole square bolos have death to doubt was good plus half of you don whole scared digital. Then we subtitled these value in this equation we have as his equal toe zero to go by when receptor told the value off Abdullah Streeter Duri's sign Square cubed feta plus nine course square Fujita, Do you know when we saw the use integrate calculated Oh, Then we use integrate calculator. You know that the value off s is equal to tartine 0.36 fight
All right. So the question we have today were asked to find the ark likes of the curve and approximate numerically if needed of the following parametric equations. So we have this. We have X equals Khost t Sorry, not coast. It's to coast t my bad there to coast e with t being the variable in the parametric equation. And our why is equal to four sign t. So what we're asked to do here is we need to find the arc light. And, um so the first thing we need to do is we need to differentiate both X and why? So we have X primes is going to be equal to the the, um, derivative of to Khowst. He is going to be negative to sign t, and then we need to find why primed and why Primes in this case is going to be equal to four coast. He remember that he is just another variable. It acts the same as any other variable. There's no special rules involved with it. Um, now we can use the formula for the are quakes of a curve which is s is equal to our, um our B and our A, um and it is thes square root of X prize squared. Plus why primed squared de of teeth. So now, risking up like everything in, uh, because this is a sign and co sign question, our range is gonna be two pi and our a is gonna be zero. It's the square root. Our ex prime does negative sign their negative to sign of tea squared plus four coasts t squared. Still all under the square root on it is the of tea. So now we just need to integrate using Ah, any method you do. Um, well, first thing that's gonna apply that square root and so that is gonna be for sign of science squared of T plus 16 close of t squared. Do you have teeth? And then once you use on a miracle method to find this and evaluate the integral do you Finally, after you should be getting is 19 0.3769 And that is the final answer to this question.