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 x^4/3. 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')