Chapter 5 begins with Integrals, which, according to DBU at least, separates Calc 1 thinking from Calc 2 thinking. I remember ending last semster with integration, though I don't remember much of the technique behind it. So, note to self: Refresh integration technique.
So we want to know the area under a curve of function. Given the vague function y=F(x), we see the area that is encased between the origin, the curve, and within given stopping points A and B. Since the sides are all curvy, it's not as quick to figure out as straight shapes like squares, triangles, etc. So we have to come up a precise, mathy definition of what an area means.
We figured out the definition of a tangent line by taking two points - the point we wanted the tangent of, and another point - and moving that second point closer and closer to our tangent point. The slope would keep approaching the slope of what the tangent line would be, so the limit of the slope was the slope of our tangent line. We're going to do pretty much the same thing to get the area under the curve, but with rectangles.
Example
We're given the curve y=x^2, and want to know the area under in from 0 to 1.
By making vertical lines from an X value every 1/4 distance (1/4, 1/2, 3/4, 1) we cut the area into four rectangles. We decided the width was going to be 1/2, and, since the length is how far from the origin to the line, the length will be f(x) for whatever point we're on.
We can extend these lines into full rectangles and add the area of all of them, which, in the example, comes out to .46875. We know that's too much, so we cut the rectangles' widths in half, doubling the number of them. We have to decide whether to use f(x) as the left end of the rectangles or the right end, which results in us either overshooting or undershooting our answer.
The smaller we make the rectangles, the closer we get to the actual area of the curve. Just looking at the numbers from the example, it looks like we keep getting closer and closer to 1/3 as our area.
Example 2
So let's see.
We're going to call Rn (there's not really a convenient way to write subscripts in basic HTML. Oh well.) the sum of all of the areas of the rectangles in our area. If n is the number of rectangles we're using, then the area of each will be (1/n)x(1/n)^2 + (1/n)(2/n)^2 since (1/n) is the width we chose, and we're increasing it by 1 and squaring it each time to get the length.
We apparently have a formula for how to summarize the sum of the squares of first n positive integers, mentioned casually as if I'm supposed to hold every formula I've seen since Algebra 1 in my head forever. I'll look that one up later, and take their word for it now. Anyway, this formula is [n(n+1)(2n+1)/6] Multiplying that by the 1/n^3 we pulled out of our series, we get [(n+1)(2n+1)/6n^2].
While taking the limit of that, we can separate it into three separate fractions: (1/6)[1+1/n][2+1/n]. If we let n go to infinity, we see the limit becomes 1/3.
Since the limit of the sum of the areas of rectangles inside this curve is 1/3, we know that the area of the space under the curve is 1/3. Yipee!
So we go back to our weird curvy thing at the beginning of the chapter. To get general principles from it, we're labeling everything with variables. So we take S, the area under the curve, and divide into n strips, which we label Sv1 (little V is a subscript now. It's like squaring^, but pointing down. Deal with it.), Sv2, all the way to Svn, and assume they will all have equal length because we don't need to make it hard on ourselves unless we're looking for something fun to present a paper on.
a and b are the endpoints of our interval, so if we're dividing it up into n equal segments, the width of these will be (b-a)/n, or (Delta)x. Since we're starting at a, the actual points of the rectangle segments (assuming we're using them as the right ends) are xv1 = a+(del)x, xv2 = a+2(del)x, etc.
So the sum of all of these rectangles will be the width(which is our delta X) times the hight (which is the value of the function at xvn.) and we substitute the values of whatever problem we're working on.
Since the estimate gets better the more rectangles we have (the higher the value of n), we can define the area under a curve as the limit of Rvn as n approaches infinity. It also works for the left endpoints, so Lvn also works. It then shows that we can say "screw endpoints!" altogether and start at any sample point, and the theorem still works. The symbol for "any sample point" is xvi with a star above the i. I wonder how long it will be before we run out of symbols completely.
Since that takes up a lot of room, we can use the sigma notation of (sigma from i=1 to n) [f(xvi)](Del)x. Much better.
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The next example just work out a problem, so I'm not summarizing that. Next is the Distance Problem.
We want to know how far something went in a given time if its velocity kept changing, like someone texting while driving. If you graphed it out, you'd get a curvy graph like the ones we just figured out the area of.
Example 4
A car's odometer is broken and the poor OCD math major who owns the car can't get it fixed yet, but has to know exactly how far he went. He's not car-savy enough to fix it himself, the shop kids still give him wedgies, but he is smart enough to get a device running to capture the exact speed he was going every five seconds.
I'm not recreating the table here, but every 5 seconds we get speed readings. Take my word on the values.
Seeing that the velocity doesn't change drastically during the 5-second intervals, we're going to pretend we were going exactly as fast as the first interval says for the whole 5 seconds. So if he went 25 ft/sec for the first five seconds, he covered 125 feet. The next interval has him going 31 ft/sec for the next 5 seconds, so that adds 155 feet. If we do this for the whole table, we end up with an estimate of 1135 feet. We can also do this using the speed at the end of the 5 second period instead of the speed at the beginning, which would give us an estimate of 1215 feet.
Lo and behold, we are actually taking equal distances on the x axis, and multiplying them by their value where they meet the graph we'd make from the points. This sounds suspiciously like what we were just doing. Sure enough, if we plot the points we figured out and make a rectangle out of the y values, it looks just like what we were doing. The area of the rectangle under a segment is roughly the distance traveled.
So we can re-write our "area-under-the-curve" formula to be a "distance-traveled-by-weird-speeds" formula by making (del)(t) instead of (del)(x)and f(tvi) instead of f(x). Since measuring the time more frequently makes smaller, and thus more, rectangles, the limit of the above equation as n approaches infinity is the distance travelled by the thing we're tracking. Or: The distance traveled is equal to the area under its velocity function.
The book assures us that the area under a curve ends up representing all kinds of fun things later on.
Meanwhile, the nerd with the broken odometer, if he managed to avoid careening into oncoming traffic while figuring this out, completely geeks out, pulls over to the side of the road, writes an Android application to do these calculations and replaces his odometer with it, resulting in him being late for the first date he'd gotten in years.
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