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Calculus and Beyond Homework Help
How to Verify Green's Theorem for a Given Rectangle?
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[QUOTE="jackarms, post: 4669576, member: 500976"] You can do it a couple different ways: One is to parameterize each segment of the rectangle and evaluate four integrals in terms of t. For example, the path from (-1, -2) to (1, -2) can be parameterized as [itex]\stackrel{\rightarrow}{r}(t) = \left\langle -1 + 2t, -2\right\rangle[/itex] where [itex]0 \leq t \leq 1[/itex], so you can have [itex]x = -1 + 2t[/itex], [itex]y = -2[/itex], [itex]dx = 2dt[/itex], and [itex]dy = 0dt[/itex]. Then just substitute everything and evaluate, then repeat the process three more times. Another way, since all the paths are straight lines either vertical or horizontal, is to evaluate the integrals without parametizing the path. Make a specific path of vertices, say the path you listed of [itex](-1,-2) \rightarrow (1,-2) \rightarrow (1,1) \rightarrow (-1,1) \rightarrow (-1, -2)[/itex], then create integrals from each of those paths. For example, for the first path [itex]x[/itex] goes from -1 to 1 and [itex]y = -2[/itex] and does not change, so the integral becomes: [itex]\int^{1}_{-1}x^{2} - 4xdx[/itex]. Either way will work. [/QUOTE]
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How to Verify Green's Theorem for a Given Rectangle?
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