Question

Find ∫C−ysin(x)dx+cos(x)dy∫C−ysin(x)dx+cos(x)dy where C is the parabola y=x2y=x2 between (0, 0) and (4, 16).

∫C−ysin(x)dx+cos(x)dy=∫C−ysin(x)dx+cos(x)dy= _____

Answer 1

To find the line integral, we need to parametrize the curve C and express the function in terms of the parameter. Then, we integrate the function over the interval of the parameter.

Step-by-step explanation:

To find the line integral of the given function over the parabola, we need to parametrize the curve C. Since C is the parabola y = x^2 between (0, 0) and (4, 16), we can parametrize it as r(t) = (t, t^2), where t goes from 0 to 4.

Now, we can express the function −ysin(x)dx+cos(x)dy in terms of t. Using the chain rule, we have:

−ysin(x)dx+cos(x)dy = −t^2sin(t)dt + cost(2t)dt = (cos(t) − t^2sin(t))dt

The integral becomes:

∫C−ysin(x)dx+cos(x)dy = ∫(cos(t) − t^2sin(t))dt

Integrating this function from 0 to 4, we get the final answer.