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Problem 18 Medium Difficulty

Integrate $f(x, y, z)=-\sqrt{x^{2}+z^{2}}$ over the circle
$$\mathbf{r}(t)=(a \cos t) \mathbf{j}+(a \sin t) \mathbf{k}, \quad 0 \leq t \leq 2 \pi$$.

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Okay, folks. So in this video, we're gonna take a look at problem number 17. Where were given a function f which I'm going to Ah. Well, actually, this is not from number 17. Does problem 80? I'm so sorry. Um, probably 18. Where were given the function F which is Ah, negative X squared plus z squared with his our function and that we want integrate this function over the circle are of tea is given by zero. Why is a co sign T and Z is a scientist e where t is going to range between zero and to Hi. OK, so that's pretty So their own. Let's do it. That's, um that's such a greatest function. We have function f is given by negative x zero, which I'm I'm just gonna leave exiled. Plus these word, um, but Z is a sign t So the squared is a score sine squared a t So we have a squared sine squared of tea. Um, so now multiplied by ds. Okay. Where ds is really just, um just dx DT squared plus d. Why did he squared close? These e d t squared multiplied by the tea. The X x zero. So the x t t zero Um, the Why did he is not zero obviously, Um, plus the Z d t squared. And these two are both non zero. So that's Ah, that's right. It out. Let's crank it out. So we have square root of base bird science grade of tea multiplied by the Yes, let's see. What do I did? He is. Well, why is a cost? So why do you Why? Why? Prime is just minus a Scient e spared. Plus Z is a science. T said these e t is just a co sign t So we have a co sign t squared the whole thing multiplied by d. D. Therefore ah, let's see what we have years old here. We can simplify this expression to a little bit because we have on the left hand side, we have a squared sine squared, and here on the right side, we have a squared cose iceberg. Now that's gonna I might bring about for you because, you know, a squared sine squared plus B squared co size worried If you pull a squared out, you get a squared multiplied by science where plus co signed square, which is one, um by there's a square root. So we're gonna do the square root of X squared, which is going to be It's you. You see the thing here I want I want to rise. I want to write this, but I can't. The reason I can't is because we're never told whether a is positive or negative. So what I'm gonna do is I'm gonna rise absolute value of a because that's, you know, algebra, therefore the same thing. Same thing here. Um I want a pole, a score science core out of the square root because the square root is way too ugly. But I can't because we're never told whether a is positive or negative. Same as science. Therefore, I want to write as a absolute value bar and then sign t Apple value bar multiplied by Ah, absolutely, Barve A and D t. Okay, that's that's what I'm gonna do. Therefore, because of the fact that the aces the constant I'm gonna pull that out of the integral. So we have Ashley value bar of ah vase scored by because of the fact that a square is definitely positive, you know, they could be negative, but a score is always gonna be positive raise. And these two, uh, absolute value bars, uh, just write it like this, uh, absolute value of sonny T d t. Where t ranges from zero 22 parts. So now I will be. Now, you see, if this thing was you know, this if he used to absolute value bars weren't here. We could literally fish the video right now, because that's all because this interview is really easy to do without the actual value bars. But because the apple value bars air here, we're gonna have to be a little bit smarter and right the integral as two parts here. Okay, well, there's really more more than one way of doing this, by the way I'm gonna do it is by recognizing that sometimes sign of to use bigger than zero and sometimes is less than zero. Um, and I'm gonna treat those two cases separately. That's why I wanna write. Asked. That's why would you write this integral as two parts? Well, let's Ah, let's think for a moment when sign is positive. Okay, well, a sign is just the y value when you're when you're talking about a unit circle sign of Fada is really just the y value of the position that you're in. Um, and these 2/4 these two quadrants quadrant one and quarter to both have why? Values of bigger than one. I mean, bigger than zero. I think that's pretty obvious. And Quadrant three in Quadrant four have wire values of a negative value song winner. So that's why I'm going to separate this integral right here into two parts. Well, the first part I'm gonna come under let t range from zero to pi. That's the first integral. And the second Integral is where is where she's going to range from pie to two pi because that's where sign becomes negative. Okay, um de t d t. Okay, um, that's food. Let's fill in the blank here. What is the blank? Well, when tees between zero and pine, that means we're in the 1st 2 quadrants and the 1st 2 quadrants is where sign is a positive sign gives you a positive value. That's why when you're right, that's why I'm gonna write scientist E as just as just scientist E. And there's no difference between absolute value of a positive number, the number itself. That's why that's why I can remove the positive, the the absolute value science. What about the second? Integral, while the second indigo is where things get tricky Because for the second World War, and we're we're in these two quadrants wanted three important for that's where sign becomes negative and when sign if he is negative, the absolute value of sign anti is just the negative of that. Because I have values, always a positive number. You know when when b, which is a number, is when When this is a negative number, absolute value. If he is just it's just negative. But I hope you know that that's really just elementary school algebra. So I'm gonna write this as a negative 70. Okay, so now we can simplify this. Ah, lot more. Okay, we can simplify this a lot more. We have negative co Scient E for the first, integral from 0 to 2 from zero to pi minus plus co sign t from pie to two pi. Okay, now let's Ah, let's see if we can crank this out. We have ah ah. Negative of a negative one minus one plus Co sign T Co. Signed two prices. One Martinez co sign pious, negative one. So that's Ah, that's a lot of one's here, minus a squared. Let's crank this up. We have negative. Ah, native to plus one plus one. This is two. So we have negative base bird. Ah, double negative science gives you a positive science that we have two plus two, which is four. So we have negative four. A swear e think, uh, that that's really a lot of numbers too quick out. But But that's it. That's it for this video. This is our finally answer. Um, this is this is the answer for problem number 18 where we have dysfunction right here. And we're integrating the vertical. That's it for this video. Thank you.