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Felicia S.

If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$

01:34

Scott N.

Kim H.

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Alrighty. So now we're trying to find the work done by this force field. So that's just a line integral on the particle moving along the boundary curve of this surface. So it's the peace of the sphere. Looks like a radius, too. Let's sketch the regional quick. So here's 12212 Okay, so that's kind of what the surface looks like. It's just the kind of part of the sphere of radius to this lying in the first octane like that. And now we're giving it kind of an upward orientation. Which means that viewed from above, if we look at the boundary circle just right here, it's oriented counterclockwise like that. Okay? And we want the work done as the particle moves along this path from there all the way around. Okay, So what we really want if this is our curve C and then our surface is s. So the work done is this line integral? Oversee the vector field in the tangent direction. But we want to use stoke serum and actually solve this integral as a surface integral. So Stokes Theorem says that this is the surface in a girl over the surface that this curve is bounding this little piece of the sphere of the curl of the vector field like that. All right, And then adding up over surface elements do sigma. So the first thing we need to do is parameter rise the surface. But this is part of a sphere. So we're going to recycle are premature ization of this fear. So recall this, uh, spear has radius too. So row is gonna be too So this should be too. And then sign s because scientific to sign s scientist E and then to co sign s But the range for S and T well, fee is just going from zero to pi over two. And data is just going from zero to pirate too. It's beer parentheses, Really? Right there. So my pears s and t are living in this rectangle in the S E s t plane. Alright, so let's get to work on setting up our surface integral. So we need to take the partial with respect s which is going to give us too coziness Guzzanti and then to coziness scientific and then negative too. Sign s and then partial with respect to t is gonna be minus two. Sign s sign T here. We'll have to right. Very easy to make mistakes with all these signs That ghost signs This should be sign. I know I was right. That's a good sign. What I say let's see. This is going to be signed us and then cousin T and this is actually going to be zero. Alrighty. Let's take the cross product to find the normal direction pointing outward. Okay, so we have zero and then minus this guy. So that'll be plus to sine squared s because I don t And then let's see, here we have zero and then times this So this will be see turns, but it's OK, so it's minus this, but the whole thing's minus so it will be too sine squared s and then it looks like 70. And then what do we have? We have, actually you know what? You should be forced 44 because two times two is four and then here we have four. Looks like we did a coastline square T Yes. So let's do this. So this is gonna be four coziness Sinus, and then we have cosign square t And this is gonna be plus four because I'm a sign a sign security. So the whole thing is gonna be before co sign s sign us. All right, So the next thing we didn't need to do is find the curl of the vector field. So in this case, the curl of F is equal to Well, let's see, I j in k. And then we have partial with respect, X partial with respect. Why partial With respect to Z, and then the component functions P Q and R or here's people Here's cute And here's our all right, so what do we get? We have partial of X here, so it's gonna be two X and then minus this partial y that zero and then minus this zero minus this, it's gonna be a to Z, and then partial effects here is going to be two x again. Um, yeah, Actually, this is gonna be too why it looks like yes to why minus zero and then to z and then two X. Yeah, that looks good. Okay, so? So we can actually evaluate this at y z and X. The various premature is ations. So this is going to be too Times two So four, this will be sign s scientific And then this will be really four co sign of s And then this will be for sign of s co sign of tea and recall that what we had from RS cross our t was this for sine squared s because on TV than for sine squared us scientific than four coziness Sign us. Alrighty. Now we just need to actually do this Stop product and take the integral So the integral is going to be from zero to pi or two for both angle both of the angles. And when we do this notice we're going to get a lot of 16 so we can factor out of 16. And then what do we get here? We get signed Cubed s then sign t co sign T plus Science squared s cosign s scientific right. And then let's see what we have here. We have sine squared s and then co Sinus society, and we'll just do DST t that is DT. Okay, so now we just have a bunch of trig intervals to solve and this really isn't so bad and the cool thing is is that this entire integral ends up just equaling one. So the final answer is just 16 on. I just for brevity. I'll just leave it kind of up to you toe work through all of this. But, you know, this is mostly just a lot of use. Substitution is like here. You wanna let you be coziness here. You wanna let you be? Sinus Here. You wanna let you be Sinus, etcetera. And the final answer is 16. So that's the work done by this force field on the particle moving around this particular boundary curve. And again, we solved this by using soak serum and evaluated the line integral along the boundary curve as a surface integral over the surface.

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