![]() K unit vector or the unit vector in the z direction. Plus b plus a cosine of s timesĬosine of t times the j unit vector- the unit factor Or the unit vectors in this orange color. Terms and the t terms in different colors. Videos on parameterizing surfaces with two parameters toįigure out how we got here. ![]() And then I'll review a littleīit of what all the terms- what the s, the t, and theĪ's and the b's represent. Several videos because it was a bit hairy. Vector-valued function of two parameters. We can parameterize a torus or a doughnut shape as a position ![]() If you are not sure why the sqrt(1 + (f'(x))²)dx part gives the arc length of a small piece of curve, you can watch this KA vid:Īnd if you want to see an example of SA of a surface of revolution, there are several on youtube, such as this: SA = ∫(circumference of slice)*(arc length of edge of slice) That is, if the surface is formed by revolving a function f(x) around the x-axis, then: But for SA, it does matter that the surface is "diagonal", so we need to understand that the SA of the edge of one slice is the ARC LENGTH of the function times the circumference of the slice. When we do the VOLUME of a solid of revolution, we just use discs or washers, since the error from "chopping off the corners" of the actual slices turns out to vanish. ![]() The idea (as with almost ALL integration concepts) is that we will slice the object into many thin slices, and then add up (integrate) an expression for the SA of each slice. The way to find the surface area (SA) is to build on the formula for finding arc length and also the ideas for finding the volume of a solid of revolution. I searched also, and I couldn't find a video on KA. ![]()
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