When I am trying to find the area between two curves below the $\mathbf{x}$-axis, should I always have the function furthest away from the x-axis as the function you are subtracting from, or should it just be the function that is on top? And when finding the area between two curves, is the answer always positive even when it's below the x-axis?
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$\begingroup$One way to think about it is to start by writing down the problem is an order-agnostic way: if your two curves are $y_1(x)$ and $y_2(x)$, you can calculate the area between them by$$\int_{x_0}^{x_1} \left|y_1(x)-y_2(x)\right|\,dx.$$
This formula is true regardless of whether $y_1 > y_2$ or vice versa, since the absolute value will take the positive distance between the $y$ coordinates in both cases.
Now, you can think about improvements that make this (correct, but difficult) integral more easy to compute. For example, you can get rid of the absolute value if you know that $y_1(x) > y_2(x)$ for all values of $x$ (note that this means that $y_1$ is greater (more positive) than $y_2$, not farther from the $x$-axis than $y_1$.
You need to be extra careful if the two curves cross somewhere. For example, if $y_1(x) > y_2(x)$ for $x\in [x_0,x_1)$ and $y_2(x) > y_1(x)$ for $x\in (x_1,x_2]$, you will need to break up your integral into two pieces in order to correctly compute the total area under the curves:
$$\int_{x_0}^{x_2} \left|y_1(x)-y_2(x)\right|\,dx = \int_{x_0}^{x_1} \left(y_1(x)-y_2(x)\right)\,dx + \int_{x_1}^{x_2} \left(y_2(x)-y_1(x)\right)\,dx.$$
$\endgroup$ $\begingroup$Welcome to MSE!
You have asked some good questions, and I will try to answer them all. I won't address what happens when using calculus to find the area between the $x$ axis and a curve when the curve crosses the $x$ axis bewteen the two relevant endpoints, unless you specifically want me to.
Firstly, when using calculus to find the area between a curve and the $x$ axis between two $x$ values , if the curve is completely below the $x$ axis then using calculus will yield a negative area. So if you wanted the actual area, ie the absolute magnitude of the answer you got just ignore the $-$ sign.
Now suppose you had $2$ curves, both (completely) below the $x$ axis between our two $x$ values we are looking at, and they don't intersect, and we want to find the area bounded by them in this region. Then the negative area can be found by finding the integral of the lowest curve bewteen the two $x$ values and then subtracting from it the integral of the upper curve bewteen the two $x$ values; this will give you the negative of the area that you are looking for. Again, if you wanted the actual area, ie the absolute magnitude of the answer you got just ignore the $-$ sign.
I hope that was helpful. If you have any further queries please don't hesitate to ask!
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