00:01
Okay, so let's compute our integral by using the parameterization of our curve.
00:10
Okay, so we want to compute the integral of f along the curve c.
00:16
Okay, this one is an integral from 0 to pi of what? f evaluated along r of t, which is sine of t.
00:31
Comma 3 the z coordinate of r comma okay here we have x which is cosine of t dot product and the r is cosine of t negative okay oh sign of t sign of t the derivative of, okay, the derivative of sign which is cosine, and here zero which is, okay, the derivative, oh, i made a mistake, negative one is here, and this one is cosine, okay, perfect, in the t.
01:36
Okay, very well.
01:37
So this guy is going to be an integral from 0 ,000, to pi of negative sine squared of t plus three cosine of t everything in the t okay so now let's compute the integral of sine square of t so negative 1 multiplied by an integral from 0 to pi of sine squared of t in the t plus 3 multiplied by an integral from 0 to pi of sine squared of t in the t plus 3 multiplied by an integral from 0 to pi of cosine of t in the t okay well this guy here is clearly going to be 0 because an antiderivative of cosine is sine so we are left with the first integral only now sine squared of t okay sine squared of t can be written as cosine of 2 t minus 1 over 2 okay so sine square of t is cosine of 2 t minus 1 over 2 so this integral here is what well we are going to have an integral and we're going to have one half multiplied by an integral from 0 to pi of cosine of 2 t in the t minus one half multiplied by an integral from 0 to pi, which is just negative pi over 2...