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Problem 32 Hard Difficulty

Use either a computer algebra system or a table of integrals to find the $ exact $ length of the arc of the curve $ y = x^{\frac{4}{3}} $ that lies between the points $ (0, 0) $ and $ (1, 1) $. If your $ CAS $ has trouble evaluating the integral, make a substitution that changes the integral into one that the $ CAS $ can evaluate.

Answer

$L=\frac{205}{128}-\frac{81 \ln (3)}{512}$

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Video Transcript

she had square. So when you read here, so we have our given, which is why is equal to X to the 4/3. We have to find the length of this curve between the 0.0 comma zero and one common one. So we were gonna integrate from 0 to 1 in respect to X. But first we find the derivative, which gives us 4/3 thanks to the 1/3 power. So l is equal to from 0 to 1 square root of one plus four x to the 1/3 power over three square D X. We're gonna substitute X. A sequel to you. Cube gives us X. The X is equal to three. You square. Do you? Phil is equal to from 0 to 1 square root of one plus for you cubed to the 1/3 power over three square time. Three you square, do you? This becomes equal to integrate from 0 to 1 square bit of one plus for you over three square times three years square, Do you? This gives us from 0 to 1. You square square root of nine plus 16 you square do you so we're allowed to use the table of in took roles. So I'm gonna use the into girl that says integral You square he square plus you square do you is equal to you Over eight times a square close to you Square times square root of a square Policy A square minus a to the fourth over Eat Helen of you plus square root of a square Tewes, you square plus c So we have this into girl. This is equal to for from 0 to 1 you square square root of nine over 16 plus you square, do you This is equal to four Sarah one you square square root, uh, three over four square plus u squared to you, which is equal to four arms you over eight three over four square plus two years square square root of three over four square plus you square minus one over eight homes three over four The fourth power I'll end of you plus square root A three over four square plus you square and we integrate this from 0 to 1. This is equal to four times you over eight 34 square plus two you square comes the square root of 34 square plus you square minus one over eight tons. 3/4 over four l and of you plus square root of three over four square. Plus he is square. We integrate this from 0 to 1 and it gives us 205 over 128 minus 81 Ln of three over 512.