7.2 Calculus of Parametric Curves

[STARTS AT 17:20]

Instructor: So let's go ahead and find the surface area of a sphere of radius R, centered at the origin. So our surface area, s. And we are using this formula because we have parametric equations. And actually we don't have any equations. So let's go ahead and write something down here. This is a sphere centered at the origin, of radius, r. So our x value is given by r cosine t, as we've seen previously. And our y value is r sine t. And in this case, because it's the surface area, we actually want to go from 0 to pi. That is, we're taking this half circle and rotating it to find the surface area. 

So our functions then, are going to be given by that. So we'll say 2 pi from 0 to pi of the square root of the derivative of x. And actually, let's add in the original function, y of t. So r sine of t. And the derivative of x is r, negative r sine t squared. There is r cosine t squared dt. 

All right. Messing with that a bit. That is going to be r squared. OK. Let's go ahead and write that in there. So r sine t. r squared sine squared t plus r squared cosine squared t. Now the r-squared can be factored out. So we're going to have an r-squared out here. Sine squared t plus cosine squared t. Integral 0 to pi, 2 pi. 

This term is one. Again Pythagorean identity. So we have r squared, which is actually a constant, does not have a value of t in it. So this is 2 pi r squared integral from 0 to pi of sine of t, which is going to be negative cosine t. evaluated from 0 to pi which is-- Well the value of pi there, of cosine is negative 1, so that will be 1 minus, evaluating that at 0. That is-- did i say that right? Negative 1 so that'd be positive 1? Yes. OK. 0 would then be 1. So that is minus 1. OK. So that would be plus 2 pi r squared, which is 4 pi r squared, which is exactly the surface area formula that we are familiar with. 

So we can actually derive some formulas based on these parametric equations. 

[ENDS AT 21:03]