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fleazo

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When we are integrating using cartesian coordinates to find the area under a curve, area under the x-axis is negative and area above the x-axis is positive. This makes sense when I think of the integral in terms of reimann sums because because we are just summing areas of rectangles using the formula f(t)*(t-a) for some t in the interval we're integrating over. If we have an f(t) under the x axis, that means f(t) is negative, and since dx is "positive", we would get a negative number for the area of the rectangle.

But when thinking of polar coordinates I'm confused. In polar coordinates, θ is like our "x" and r is like our "y". So it seems the analogue to this situation would be when r is negative. But when thinking of this in terms of reimann sums, that doesn't seem like the case. Since the area of a sector of a circle is (1/2)r

^{2}θ, if r is negative it just becomes positive when we square it. So the only thing that would make this amount negative is if θ is negative. in the integral, dθ takes the place of θ in this equation. Does dθ ever become "negative"? Do we have to worry about negative area when dealing with integrating using polar coordinates?