~AREA UNDER A CURVE~

$astronaut$

~The area under a curve & between two x values may or may not exist (see link, LUB & GLB, under enrichment topics). In a basic course in calculus, it usually does.

~We first get a good sketch of the curve in question & the area that we would like to find. Let x=a be the x value for the left boundary & x=b be the value on the right boundary.

~In the beginning stages, we approximate the area by partitioning the interval, [a;b] into subintervals with widths of delta x for each. In actuality, the widths could be different sizes, however, having them the same size makes things simpler. These widths serve as the bases of rectangles drawn up to the curve. We can either inscribe (use the smallest y value over each subinterval) or circumscribe (use the largest y value) these rectangles.

~We then calculate the areas of each and take their sum.

~This will be an approximation to the area that we seek.

~The smaller the delta x (width of each rectangle), the more rectangles we are inserting, and the better the approximation.

~In the limit, as the number of rectangles approach infinity, their sum will converge to the actual area.

~This process could be quite lengthy & almost impossible to do with complicated curves. Mathematical Induction (see link in enrichment section) is often used to establish formulas to find values of special sums.

~After the fundamental tie between area & the definite integral has been established, we can find this area simply by evaluating a definite integral. I will discuss the Definite Integral in another session.

~So, to find the area under a curve & between two x values (I will assume the curve is positive), just integrate the y value [or f(x) ], then substitute x=b into the result and subtract the same thing with x=a substituted into the result. The constant of integration is subtracted away & we don't have to write it.

This is called, THE DEFINITE INTEGRAL of f(x) from a to b and is written as:

                 ∫_a^{^b} f(x)dx

~If the curve is both positive & negative between a and b, the definite integral could still be calculated, however, the result is a number that is the sum of positives (regions above) with negatives (regions below) and does not represent the area. For example, the definite integral of f(x)=sin(x) from x=0 to x=2p would give zero, even though there is plenty of area trapped. What happen? The positive region between 0 and p was combined with the negative region between p and 2p. Since they had the same amount of area trapped, their areas were the same but their signs were different, thus a zero result.

~Here are some examples to look at:

~Ex:   Find the area under f(x)=x³ from x=0 to x=2.

~Solution:     You should sketch this curve to see the region you are after.
Why? If the curve dips below the x axis between our x values that will give us an erroneous answer. (Not the case here)
Now, simply find ∫₀^² x³dx^{This is a 2 step procedure. First, integrate x³ wrt x to get (x⁴)/4 (no constant). Then sub in 2 (upper limit of integration) minus sub in 0 (lower limit of
integration).

So, we would have, (2⁴)/4   -   (0⁴)/4 = 4 sq. units.

~Ex: Find the area bounded by the parabola y = -x²+x+2 and the x axis.

~Solution: We definitely need a sketch here since we need to find the limits of integration. These will be the x values where
the parabola intersects the x axis (the zeroes of this function).

Setting y=0 & solving for x will do it: -x²+x+2 =0. Multiply both sides by -1 & factor the left side: (x+1)(x-2)=0.

Setting each factor = 0, we get x= -1 and x= 2. We now can find the area by the following definite integral:

Area=∫_-1^{² (-x²+x+2)dx}^{=[-(x³)/3 + x²/2 +2x] from -1 to 2. Sub in the limits
of integration & you should have the answer: (upper limit first)=(-8/3 +2 +4) - (1/3 +1/2 -2) = 9/2 sq. units.

~These are simple examples.
Not all problems are this simple. The procedure outline in this discussion is
the same no matter how complex the situation, however, the problem could involve
much sketch work & equation solving. So, the better you are at those, the
less frustrated you will be.

~In Statistics, the area under special types of curves take on a very
special meaning. If the curve is non-negative (never below the x-axis)
& the total area under the curve & above the x-axis is exactly
equal to one, it's called a PROBABILITY DENSITY FUNCTION.

~They define certain types of distributions where probabilities of
values in data sets (on the x-axis) can be computed. These
probabilities are nothing more than the areas under these curves.

~Some are very complicated (depending on the type of distribution). The
most popular one is the NORMAL CURVE that has a very complicated
equation & it takes multi-variable calculus to prove that the area
under this curve is one.

~There are many normal curves (different means & standard
deviations), so, to work with them in statistics, we convert them to
the STANDARD NORMAL CURVE (mean is zero & standard deviation is
one).

~We can then solve many problems in statistics involving probabilities & also play an important role in hypothesis testing.

~A few other popular probability density functions are the BINOMIAL
DISTRIBUTION (since this distribution is DISCRETE not CONTINUOUS, the
normal distribution (CONTINUOUS) is used to approximate it, given
certain conditions hold) (see topics of interest under statistics on
this site) (also a discussion on DISCRETE & CONTINUOUS random
variables is also listed), the STUDENTS t DISTRIBUTION, & the
CHI-SQUARE DISTRIBUTION. There are others which are used in advanced
phases of statistics

~See the links under topics of interest on this website (under statistics).}}