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I have a question regarding finding the area between two curves. I will link to a well known example that seems to show up in every calculus text book!

http://tutorial.math.lamar.edu/Classes/CalcI/AreaBetweenCurves.aspx

In particular on that page, I am referencing Example 6. And in particular, the page evaluates this integral with the functions in terms of x first, and then at the end of the example, with the function in terms of y. My issue is when the area is being found between the two functions in terms of y. I just don't understand how this is giving us the true area, as a lot of this area falls on the left side of the y axis, so won't the area be "negative" there? i mean even just look at that example and see the line x=y+1. From the region of y=-1 to y=-2, the x values are negative. So say we were looking at the integral of this line from y = -2 to y= 4. What we are really getting is the area between the line and the y axis from y = -1 to y = 4 and then subtracting off the area between the line and the y axis from y = -1 to y =-2.

Do you see where I'm going with this? Why does just taking the integral of f(x)-g(x) give us the actual shaded area vs just the "net area"? If we want the actual shaded area, why don't we take the integral of the "right function" from -1 to 2.5, then the integral of the "right function" minus the "left function" from 2.5 to 4, then the absolute value of the integral of the "left function" from -1 to 2.5, and the absolute value of the integral of the "left function" minus the "right function" from -1 to -2?

By the way I know I'm wrong I'm not at all trying to say I'm right, I just want to understand what I'm not understanding

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# Area between two curves problem

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