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What we will attempt to start to do in this video is to take the surface integral

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of the function x squared over our surface where the surface

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in question--the surface we're going to care about is going

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to be the unit sphere, so it could be defined by x squared

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plus y squared plus z squared is equal to 1

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what we're going to focus on in this first video, because it will take us several videos to do it

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is just the parameterization of this surface area right here

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as you'll see this is often the hardest part

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because it takes a little bit of visualization

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and then after that it's just kind of mechanical,

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but it can be kind of hairy at the same time so it's worth going through

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So first let's just think about how we can parameterize

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and I have trouble even saying the word,

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how we can parameterize this unit sphere as a function of

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two parameters

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so let's think about it, let's think about it a little bit

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so first let's just think about, let's just think about

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the unit sphere

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i'm gonna take a side view of the unit sphere,

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so let's take the unit sphere, so this right over here is

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our z axis

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That's our z axis, and then over here I'm gonna draw

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This is going to be not just the x or the y axis,

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this is going to be our entire xy plane viewed from the side

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That is the xy plane.

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Now, our sphere, our sphere, our unit sphere might look

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something like this

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The unit sphere itself is not too hard to visualize

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It might look something like that

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The radius at any point is 1

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That length right over there is 1,

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and this is a sphere, not just a circle

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So I could even shade it in a little bit just to make it

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clear that this thing has some dimensionality to it

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so that's shading it in it just makes it kind of look a

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little more spherical.

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Now let's just attempt to parameterize it, and the first

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step, let's just think, if we didn't have to think above and

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below the xy plane, if we just thought about where this

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unit sphere intersected the xy plane, how we could

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parameterize that.

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So let's just think about where it intersects the xy plane

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it intersects it there, and there, and actually everywhere

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it instersects it everywhere.