Find the value of the line integral along the curve C of the function 4xdx plus zdy plus 4y squared dz, where C is the curve parameterized by r of t equal to angle bracket 4 cosine, open parentheses, 2t, close parentheses, comma, 2 times sine open parentheses 2t close parentheses, comma 3, where 0 is less than or equal to t, is less than or equal to pi over 4.

First notice that r of t gives us values for x, y, and z for the 3 components respectively. And also we can determine what dx, dy, and dz are by finding r prime of t. I'm going to go ahead and write this on the side. R prime of t equals the derivative of 4 cosine of 2t, quantity.

So this will be the vector negative 8 sine parentheses 2t close parentheses, comma, 4 co-sign, open parentheses, 2t, close parentheses, comma, 0. End angle bracket. Now what substitute these values into are integral, this will be the integral from 0 to pi over 4, all with respect to t, and our function will be 4 open parentheses, 4 cosine, open parentheses, 2t, close parentheses. Close parentheses, multiplied by negative 8, sine, open parentheses, 2t, close parentheses, plus 3 times 4 cosine open parentheses 2t close parentheses plus 4 open parentheses 2 sine open parentheses 2t close parentheses close parentheses squared times 0.

Now we might combine some things and simplify for you. And notice, this is the integral from 0 to pi over 4 with respect to t of the function negative 128 sine open parentheses 2t close parentheses cosine open parentheses 2t close parentheses. Plus 12 cosine open parentheses 2t close parentheses.

Now, applying u substitution, we can note that this is going to be equal to the antiderivative negative 32 sine squared parentheses 2t close parentheses plus 6 sine open parentheses 2t close parentheses, evaluated from 0 to pi over 4, which is equal to negative 26.