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Problem 29 Easy Difficulty

Find the exact value of $\int_{C} x^{3} y^{5} d s,$ where $C$ is the part of
the astroid $x=\cos ^{3} t, y=\sin ^{3} t$ in the first quadrant.

Answer

$\frac{945}{16,777,216} \pi$

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Video Transcript

{'transcript': "Hi. We're gonna be solving for this integral in this problem. And so first thing we have to do is we're going to do a U substitution. So we're going to say, Are you is equal Toothy sign of the X. And now we have to find a driven of both sides with a D u as equal to three coasts. Three x d x using the chain rule. Ah, we found that. And now what we're gonna do is we're gonna divide, um both sides by three Khowst, three acts. So we're going to have d'you over three. Khost three x is equal to D X. And from here, we're going to substitute in for our sign three acts that we're gonna substitute in for RDX. So we're going to have the integral of you to the fifth. Um, Times coasts three x times d you over three. Khost reacts, and from here we can do is cross out our coast three x and we can take out our one third. And so we're gonna have the one third times you to the fifth D you And from here, we're going to do is gonna use the equation that says that X to the n is equal to X to the n plus one over and plus one on DSO From here, we're going to have ah, that the integral of you to the fifth, do you Time from third is equal to one third times you to the six over six plus See on. So from here, all we have to do is plug back in for are you and are you? Is equal to the sign of three X So we're going to have it's one third times be sign of three acts to the sixth over six plus C, which is equal to the sign of three x six over 18 us see, and that is going to be our final answer."}

George Washington University, The
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