While subject to the force F(x,y) = <?x + y^3, x^2 + ?y>, a...

While subject to the force F(x,y) = <?x + y^3, x^2 + ?y>, a particle travels once around the curve y = sin x from (0,0) to (?, 0), and the line segment from (?,0) to (0,0). Use Green's Theorem to find the work done by F.

Transcript

00:01 So for these problems we are given a curve.

00:03 Let us represent the curve which is around the function or curve y equals to sine of x from the origin zero zero to the point pi pi comma zero and then there's a second segment to that the line that starts at the point five pi comma zero and ends at the origin so at the curve c is going to be a complete closed curve let us try to see if we can illustrate this by sketching this curve on the x and y plane so here we have the y axis there is the x axis and let us sketch the graph of y cosine effects which starts at the origin goes up and it comes back down to the x axis as such and then it continues on to oscillate after that so here we have the origin at 0 -0 then it goes around the curve which is y cosine of x comes back to the x axis at the point pi comma 0 and then it takes another segment along the x -axis the curve c starts at pi comma 0 and then comes back around to the origin the entire curve bounds a region let's call that region r the vector field that we are given here f is equal to the square root of x plus y to the third powered for the first component and for the second component is going to be x squared plus the square root of y so if we are finding the work the integral along c of f dotted with d r this is representing the work done by the vector field along the curve this by green's theorem is going to equal to the double integral over the region r the region bounded by the curve c of partial partial x of the second components of the vector field that is x squared plus the square root of y minus partial partial y of the first component of the vector field which is the square root of x plus y to the power of three and then d a at the end next this is going to equal to the integral from zero to pi however notice that the curve instead of going counterclockwise is it goes clockwise so before we continue on, that's a good observation to make, that we have to multiply this by a factor of negative 1, because this formula would have been positive if the curve had a counterclockwise orientation.

02:53 But because the curve goes clockwise, we have to add that additional negative sign...

Question

While subject to the force F(x,y) = <?x + y^3, x^2 + ?y>, a particle travels once around the curve y = sin x from (0,0) to (?, 0), and the line segment from (?,0) to (0,0). Use Green's Theorem to find the work done by F.

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