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

$$\int_{C} x y^{4} d s, \quad C \text { is the right half of the circle } x^{2}+y^{2}=16$$

$\int_{C} x y^{4} d s=\frac{8192}{5} \approx 1638.4$

Vector Calculus

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So in this video are asked to determine the line to grow of X times y to the power board ts. All right, we're given that c is the right half of X square. Plus Ally Square was 16 and then we're supposed super of amateur rises ourselves. But this isn't simple because X squared plus y squared is equal to 16 is basically just a circle of radius for but the more asked only to determine the right half. So this is why right over here, we only have the right half of the draw, and we know how to grab it dries, um, circles. So we know that access just barcos I Inti and lines are 70 but our are in this case is just four reduces for and then our t is gonna go from negative pi over two. So we start from this point and we go all the way up to pi over to this. So this is our parameter organization, all right. And then we know the second thing that we know is that yes, we need to find the Ark link and we know that DS is just square boot of d y by D Squared plus the expert 80 square T T. And riveted off with respect to so d explain TT's with derivative of four co scientist er's negative for scientific and they give a beauty of a forest lane. He's just for coast. So I get this right here. All right, I know we expand this. We get 16 Science Square for 16 co sites where what we can do is pull out 16 is calling factor, and we're left with Science Square Costco size. Now sign square coastline square, just one. So we're left with the square root of 16 DT, which is just for some we found Yes. Okay, now the next step this we're gonna plug everything back in. So then plug in our X and Y's and DS, and we're gonna write everything in terms of teeth. So here we have a four here we have a fork in the power for and here we have four. So that's just four to the power of six, which we can pull it to the sign and then we have the integral from negative by over to defy were to sign t to the power for co sign TV. All right. Now four to the power of six is 4096. And then now again, we're gonna try to use a far you substitution. We're gonna you equal scientific. And when we do that, our d'you is just coastline TDD. So what we have here is Go sign. So what we have right over here is just Do you? So you did a power for, do you? And now we're gonna try to write everything in terms of you. So instead of negative pi over two, that's our tea. We're gonna plug that into scientist, we get negative one, and then instead of pi over to our upper limit, we're gonna plug it in. We're gonna write it in terms of you. So that's just signed off pile, which is one. Finally, we get this rather simple into go over here and the integral of U to the power. Or do you use just U to the power five divided by five? All right, that we're gonna Our upper limit is one are lower limit is negative. One. It only subtract thes. We get 4096 times two divided by five. Just just 8192 divided by. So you can leave it like this or write it as a decimal. But this is our solution. This is like