Tuesday, September 15, 2009

Calculus in 5 minutes (part 4)

If you haven't read the earlier posts, they are here, here and here.

Now that was interesting, you say. But how about a little proof ? For a mathematician, the word "proof" is like waving the red handkerchief to a bull at Pamplona, but I'm not going there. However, it is always a good idea to check things a bit to see if they make sense. Consider the simple equation:

y = x2

We want to know the area under the curve.



I know how to do that, you say. Remembering that the curve itself is the derivative of the area function f(x), we have:

y' = x2
y = 1/3 x3

evaluated between x = 1 and x = 0:

area = 1/3


That was way too easy. But consider this: how about the area above the curve?



Notice that we have

y = x2
x = √y = y1/2


I get a little freaked out seeing x as a function of y, so I am going to mentally turn the curve through 90 degrees and then make its mirror image reflection. I get this:



Now, our differentiation trick works for fractional exponents. So we have:

y' = x1/2
y = 2/3 x3/2

evaluated between x = 1 and x = 0:

area = 2/3


And, as we might hope and expect, 2/3 + 1/3 does equal the area of the unit square. And note that it will work for any power of x. If, for example, we start with the curve y = x3, then the first area is 1/4 x4, and the second area is 3/4 x4/3. It's easy to see that it works for any rational power of x. Suppose we have:

y' = xa/b
the exponent is a/b, let
a/b + 1 = c/b
y = b/c xc/b

x' = yb/a
b/a + 1 = d/a
x = a/d yd/a

From above:
a/b + 1 = c/b
a + b = c

Also from above:
b/a + 1 = d/a
a + b = d

So:
c = d

a + b = c
a/c + b/c = 1
a/d + b/c = 1


As we guessed, we see that b/c + a/d = 1.

(Of course, it helps that we chose the upper limit for measuring the area at x = 1, y = 1. Evaluation for some other bound would involve a bit more calculation).

Ladies and gentlemen, will you please…give it up for Mr. Leibnitz and Mr. Newton.

[Update: blogger messed with my code. Think I fixed it:]

R code:
f <- function(x) { return (x**2) } 
plot (f,0,1,cex=3)
xvals = seq(0,1,length=1000)
yvals = f(xvals)
x = c(xvals,rev(xvals))
y = c(rep(0,1000),rev(yvals)) polygon(x,y,col='gray')

f <- function(x) { return (x**2) } plot (f,0,1,cex=3)
xvals = seq(0,1,length=1000)
yvals = f(xvals)
x = c(xvals,rev(xvals))
y = c(rep(1,1000),rev(yvals))
polygon(x,y,col='gray')

f <- function(x) { return (x**0.5) }
plot (f,0,1,cex=3)
xvals = seq(0,1,length=1000)
yvals = f(xvals)
x = c(xvals,rev(xvals))
y = c(rep(0,1000),rev(yvals))
polygon(x,y,col='gray')

No comments: