Hi! New here...I've been struggling with this problem for a couple of days. I've tried applying several examples, but I don't see where I'm going wrong. Here is the exact problem text:
By making an appropriate choice for the functions P(x,y) and Q(x,y) that appear in Green's Theorem in a plane, show that the integral of x-y over the upper half of the unit circle centered on the origin has the value -2/3. Show the same result by direct integration in Cartesian coordinates.
Ok, so let's call the Green's Theorem part, part (a) and the direct integration part, part (b).
For part (a), I really don't know how to pick P and Q. It seems I have this function, f(x,y) = x-y. And Green's theorem starts with a vector. Is my vector F=xi - yj? I also thought about taking the gradient of f(x,y), which is just =i - j and I'm not really sure what that does for me.
For part (b), I've applied many examples. Most use parametrization and set x=cos t and y = sin t. The magnitude of this is just 1, so then my integral becomes cos t -sin t dt? But then I just get 0. I have uploaded a pic of my work for this.
Any insight is appreciated! Thanks!
By making an appropriate choice for the functions P(x,y) and Q(x,y) that appear in Green's Theorem in a plane, show that the integral of x-y over the upper half of the unit circle centered on the origin has the value -2/3. Show the same result by direct integration in Cartesian coordinates.
Ok, so let's call the Green's Theorem part, part (a) and the direct integration part, part (b).
For part (a), I really don't know how to pick P and Q. It seems I have this function, f(x,y) = x-y. And Green's theorem starts with a vector. Is my vector F=xi - yj? I also thought about taking the gradient of f(x,y), which is just =i - j and I'm not really sure what that does for me.
For part (b), I've applied many examples. Most use parametrization and set x=cos t and y = sin t. The magnitude of this is just 1, so then my integral becomes cos t -sin t dt? But then I just get 0. I have uploaded a pic of my work for this.
Any insight is appreciated! Thanks!
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