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How to Integrate line bulletin

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Professor Dave here, I want to tell you about line integrals. By now we are very familiar with ordinary integrals. Integrating f(x)dx lets us find the area under a curve given by the function f(x). Integrating f(x,y)dxdy lets us find the volume under a surface given by the function f(x,y). Now with line integrals, we can integrate a surface f(x,y) along the path of some curve C. We will be integrating along small segments of the curve C, which we will call ds. Just like when we initially learned integration, these segments will be our widths, while the surface f(x,y) will be our heights. So once again, our integration will give an area, but now we are finding the area under a surface along a particular path within that surface. The way we will write this out is the integration along C of f(x,y)ds. Recall that when we learned how to evaluate multiple integrals, the x and y variables were treated independently during the integration. But now with line integrals we have a single integral

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