INSTRUCTOR: Calculate the mass of a spring in the shape of a helix parameterized by r of t equal to, angle bracket, cosine t, sine t, t with 0 less than or equal to t, less than or equal to 6 pi with a density function given by a rho of x, y, z, equal to x plus y plus z kilograms per meter.

The mass is going to be equal to the integral from 0 to 6 pi of rho multiplied by the magnitude of r prime, our parameterization, with respect to t. So let's first go ahead and find what r prime is.

So the derivative of our parameterization is equal to, angle bracket, negative sine t, cosine t, 1 and the magnitude of r prime of t is equal to the square root of negative sine t squared, which would be sine squared t, plus cosine squared t, plus 1 squared.

Now because sine squared plus cosine squared is equal to 1, this is equal to the square root of 2. Now let's replace x, y, and z with cosine t, sine t, and t, respectively, so that our integral for our mass, m, equals integral from 0 to 6 pi-- with respect to t of the function cosine t, plus sine t, plus t.

Now finding the antiderivative of this and evaluating, this is going to be equal to the square root of 2 times the function sine t, minus cosine t, plus t squared over 2. Evaluate it from 0 to 6 pi. And evaluating this, we find the mass is equal to 18 square root of 2, times pi squared kilograms.