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Evaluate the line integral, where $C$ is the given curve.

$$\int_{C} y^{3} d s, \quad C : x=t^{3}, y=t, 0 \leqslant t \leqslant 2$$

$\int_{C} y^{3} d s=\frac{145 \sqrt{145}-1}{54} \approx 32.3154$

Vector Calculus

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All right, so and this question, we were asked to find the lining to grow, and we're given that our curve C That is why I caused T X equals City Cube s t goes from 0 to 2. So it's given Pyramid very metric. Lee. First thing we have to do is determine the Ark link over DS. And then we know our formula that big ass is simply the square root of the derivative of exit respected T Square was the derivative of why would perspective squared times detail. So if we look at our X, we want to play its derivative with respect titties. Derivative cube with respected t just three t square and then the derivative a fly with respected TVO wise just t The derivative of tea is one that's one square. So the S is just a square root of three t squared all that swear plus one square spirit. So this is simply way expand is just the square root of 92. The problem for us one and indescribable of that times DT. So we determined power two years. All right, so now we're going to write everything in terms of tea So there y que burner. So we know the why is t So if we put that back in, we're going to get T Q And then instead of the yes, we're gonna put this whole thing that we just turn All right, So and then now when we plug that in So that should be it to Sorry. Sonali plant that in we This is the new integral that we must turn. All right, so now we learn we used the old tricks are the old techniques that we already know. So we have a t to the power for all the inside and a tiki don t on the outside. So what we're gonna do is we're gonna, uh that you So right over here, we're gonna do you equal 9 to 9 times teeter the power for a plus one. And if we differentiate that we get D'You is 30 60 cube pt All right now. So that means that reduce of every evil that we divide by 36 on both sides We get that que dtz quarter Do you divided by 36? So that simply means instead of what's on the inside, we're gonna put you and then teak you DT of these tool right over here. We're going to get in and see you divided by 36 then we're gonna change the limits of the integral. So remember, were given that equals zero and T equals two. So, for the for the lower limit, we're going to do nine times, uh, zero to the power four plus one. The upper limits gonna be nine times two to the power four plus one, and we get 1/36 square root of you. All right, so now we can pull that 1/36 to the outside, and then we get our limits Are new limits are now one and 145. Then we need to take the square root of the derivative. Sorry, the integral off square root of you. Do you all this? We're route of you, Do you? The integral of that is just due to the power of 3/2, uh, times two over. So we get our So this is just a regular simple into growth. And then now we plug in our limits, which is just 145 into the power of 3/2, minus one to the power 3/2 and also one divided by 36. Statue to over three. That's just one divided by or then if we work out the math, we do the algebra. My final answer is gonna be one divided by 45 times square root of 40 times, 145 times square root of 100 and 45 minus one. So this is our integral. This is our answer.