What Im confused with is why we dont have to calculate the absolute value of any areas that cross the y-axis? I kow you must break areas into parts that go below the x axis when integrating with respect to x, but why not in respect y? Ive seen areas between functions cross the x AND y axis but no change to the integration method. I am a bio major taking calculus II and i get caught up on small details like this for some reason.
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It doesn't matter whether you integrate with respect to x or with respect to y. You do not have to worry about the absolute value as long as you are integrating g(x)-f(x) or g(y)-f(y) where g is the function on top or to the right and f is the function on bottom or to the left. If g is only on top (or to the right) part of the time, then you have to break the integral up into multiple parts so that you are always integrate (top function - bottom function) or (right function - left function).
If you think about integrating with respect to x of a single function and you are using the absolute values to find the area, what you are really doing is integrating (f(x)-0) when the function is above the x-axis, and (0-f(x)) when the function is below the x-axis.
If you think about integrating with respect to x of a single function and you are using the absolute values to find the area, what you are really doing is integrating (f(x)-0) when the function is above the x-axis, and (0-f(x)) when the function is below the x-axis.