6.1 - Areas Between Curves

There was actually a section before this, but it was basically an entire chapter spent saying "If the integral is weird, substitute parts for u until it makes sense." We skipped over the reading and just did problems from it in class, so I'm not going to go through the whole chapter for now. Maybe some other time if I'm bored and in a calculus mood. So I'm moving on to 6.1.

We've already learned how to find the area under a curve on a graph, so now we're moving on to putting two lines on a graph at the same time and finding the area between them in a given interval. If we estimate it, we can use the same rectangle trick we did for the area under the curve. The only difference is that, instead of f(xi) being the height, the height will be (Assuming f(x)>g(x) between our chosen a and b) f(xi)-g(xi) since that will be the distance between the two lines on the given point. So our approximation would be Σ[f(xi)-g(xi)]▲x.

So, just like with the area under the curve, if we choose n as our number of rectangles, then the area between these two curves is the limit of our approximation as n approaches infinity. Since that limit is the definite integral of f - g, then we can define the area as A = ⌠(a to b)[f(x) - g(x)]dx.
The book also feels it necessary to point out that, if g(x)=0, then we use the old definition, which makes sense because using g(x) as the 0 line changes nothing.

If two lines are given that intersect at two different points, and your a and b are not given, then they can be found by solving their equations. If you have funky equations that would take ages to do by hand, then a graphic calculator or computer program (which I need to look into getting) will work. We can use Newton's Method, a rootfinder, or the super scientific solution of zooming in really close and seeing where it intersects.

For application-minded folk like me, the area between velocity curves is the distance between the cars at a given point, which is pretty cool.
If you get twisty curves where the lines switch from top to bottom and you a get a DNA-looking twist, then you split the area into multiple regions, find them each individually, then add them together. We can also write a slightly modified area equation that say the area between two curves is the integral of the absolute value of f(x)-g(x).

Also, if the two lines are vertical, you can treat as a function of y, which results in the same process, but flipping the whole thing sideways.

And that's pretty much it. Not a terribly complicated chapter, since once I put the work in to understand the whole integral concept, it's pretty much just application and slightly different ways of thinking about it.