Lets say we have the following graphs:
Why can we assume $f(x) - g(x)$ for volume and area, even when part of the graph is below the $x$-axis (or will go below the $x$-axis when calculating volume?) I understand how to take the area and volume and all.. but I don't have an intuitive understanding of it yet. Can this be applied to every graph problem to calculate area and volume? Please explain what happens.
I assume: ( $f(x)>= g(x) \text{ for } [a ,b]$ )
Highly appreciated,
_Bowser
The idea for the area calculation is basically the same as Riemann integration: you cover (or inscribe) the region with many narrow rectangles, and add up the areas.
The area of each small rectangle is the product of its height and its width. You get the width the same as for Riemann integration; the height is the amount you have to add to the $y$-coordinate at the bottom of the rectangle in order to reach the $y$-coordinate at the top of the rectangle: the $y$-value at the top minus the $y$-value at the bottom.
As long as $f(x) \geq g(x)$ everywhere in your region of integration, the $y$-value at the top of each rectangle is $f(x)$ at some $x$ and the $y$-value at the bottom of each rectangle is $g(x)$ at the same $x$, so the height is $f(x) - g(x)$.
All of this is completely independent of the signs of $f(x)$ and $g(x)$. For example, $y_1 = -5$ is $2$ units above $y_0 = -7$. And indeed, if $f(x) = -5$ and $g(x) = -7$, $$f(x) - g(x) = (-5) - (-7) = -5 + 7 = 2,$$ which is exactly the height of a rectangle with its top at $y_1 = -5$ and its bottom at $y_0 = -7$.
But this is not independent of the sign of $f(x) - g(x)$. Whenever $f(x) - g(x) < 0$ (that is, when $f(x) < g(x)$), the Riemann integral will include a negative term for the rectangle constructed at that value of $x$. Instead of measuring the area between the two curves in the ordinary geometric sense, the Riemann integral of $f(x) - g(x)$ over an interval in which $f(x) < g(x)$ will give you the "negative" of the geometric area.
In fact this is exactly the same thing that happens when you integrate a function that is sometimes negative: as long as the function $f(x)$ is non-negative in the region of integration, the area under $f(x)$ and above the $x$-axis is the area between the functions $f(x)$ and $g(x)$ where $g(x)=0$ for all $x$. But whenever the function $f(x)$ is negative, you find yourself in the case $f(x) - g(x) < 0$, and that's when the Riemann integral gives you "negative" areas.
If you really want to measure the geometric area between two curves, not the kind of "area" where some regions can have "negative" area and the areas of some regions can partially cancel others, you should integrate the absolute value of the function difference, $\lvert f(x) - g(x) \rvert$.