Area under the curve for 1/x

[hellosoupy]

Hi!

This is a general question that came up for me while studying microeconomics: Is there any way to find the area under the curve for the function 1/x from say, 0 to 1, since the integral of 1/x is ln x + C (the natural log of the absolute value of x that is)? I evaluated the definite integral from various tiny numbers to 1, and the tinier you go, the higher and higher the number gets.

Thanks in advance!

Jaux

 

[lex]

There is no finite area there; as you rightly point out, the integral from 1/n (say) to 1, tends to infinity.
It is interesting that the improper integral of [MATH]y=\frac{1}{x^\frac{1}{2}}[/MATH] (red line) between 0 and 1 does exist. There is a finite area.

[MATH]\frac{1}{x}[/MATH] (black line) goes to infinity 'more quickly' than [MATH]\frac{1}{x^\frac{1}{2}}[/MATH]

 

[nasi112]

There is no finite area there; as you rightly point out, the integral from 1/n (say) to 1, tends to infinity.
It is interesting that the improper integral of [MATH]y=\frac{1}{x^\frac{1}{2}}[/MATH] (red line) between 0 and 1 does exist. There is a finite area.
View attachment 27354
[MATH]\frac{1}{x}[/MATH] (black line) goes to infinity 'more quickly' than [MATH]\frac{1}{x^\frac{1}{2}}[/MATH]

The interesting thing about both of them is that as [MATH]x[/MATH] goes to [MATH]0[/MATH], [MATH]y[/MATH] goes to [MATH]\infty[/MATH]
but their area is a different story. you can see that by a little integration

[MATH]\int_0^1 \frac{1}{x} \ dx = \lim_{a\to 0^{+}} \int_a^1 \frac{1}{x} \ dx = \lim_{a\to 0^{+}} \ln 1 - \ln a = 0 - (-\infty) = \infty[/MATH]
While

[MATH]\int_0^1 \frac{1}{x^{\frac{1}{2}}} \ dx = \lim_{a\to 0^{+}} \int_a^1 \frac{1}{x^{\frac{1}{2}}} \ dx = \lim_{a\to 0^{+}} 2\sqrt{1} - 2\sqrt{a} = 2[/MATH]

 

[lex]

The interesting thing about both of them is that as [MATH]x[/MATH] goes to [MATH]0[/MATH], [MATH]y[/MATH] goes to [MATH]\infty[/MATH]
but their area is a different story. you can see that by a little integration

[MATH]\int_0^1 \frac{1}{x} \ dx = \lim_{a\to 0^{+}} \int_a^1 \frac{1}{x} \ dx = \lim_{a\to 0^{+}} \ln 1 - \ln a = 0 - (-\infty) = \infty[/MATH]
While

[MATH]\int_0^1 \frac{1}{x^{\frac{1}{2}}} \ dx = \lim_{a\to 0^{+}} \int_a^1 \frac{1}{x^{\frac{1}{2}}} \ dx = \lim_{a\to 0^{+}} 2\sqrt{1} - 2\sqrt{a} = 2[/MATH]

Indeed.

I would recommend:

[MATH]\int_a^1 \frac{1}{x} \ dx = \ln 1 - \ln a = - \ln a \hspace2ex[/MATH] ([MATH]a>0[/MATH])

[MATH] \lim_{a\to 0^{+}} \int_a^1 \frac{1}{x} \ dx = \lim_{a\to 0^{+}} (- \ln a) = \infty [/MATH]
rather than doing arithmetic with [MATH]\infty[/MATH]

While

[MATH]\int_0^1 \frac{1}{x^{\frac{1}{2}}} \ dx = \lim_{a\to 0^{+}} \int_a^1 \frac{1}{x^{\frac{1}{2}}} \ dx = \lim_{a\to 0^{+}} (2\sqrt{1} - 2\sqrt{a}) = 2[/MATH]

 

[Steven G]

This reminds me of Gabriel's horn. Although, it is the right endpoint (infinity) that is of interest.,

 

[lex]

This reminds me of Gabriel's horn. Although, it is the right endpoint (infinity) that is of interest.,

Gabriel's horn is interesting. Thanks for the reference.

 

[JeffM]

So there is a slight problem with how microeconomics is presented. It is generally assumed that the functions in economics are at least twice differentiable. But in fact the domain for economic variables is that of rational numbers and the functions are not even continuous. No one ever bought pi pigs. The number of pigs exchanged is always a non-negative integer. And if no pig is exchanged, what is the price or exchange ratio? This is usually just pettyfogging: the number of kernels of corn sold by a farmer is so large that we really are not concerned with any inaccuracy related to thousandths of a kernel.

But sometimes, especially when dealing with numbers close to 1, working with real numbers and differentiable functions simply becomes economically absurd, and we must deal with discrete mathematics to avoid idiocy. So thinking about specific functions without reference to the economic situation to which the functions are supposed to apply may create spurious mathematical conundrums. If zero is not feasible in terms of the model (and obviously 1/0 does not correspond to any actual economic situation), just define your domain so that zero is excluded.

 

[lex]

This is a general question that came up for me while studying microeconomics.

It was an interesting question and led to varied responses - we touched on calculus, Gabriel's Horn and pork bellies!