# Integration - Accurate or Approximate

We can calculate area or volume using integration. Though we divide the irregular shape into infinitesimal rectangles, I think it may not be possible in reality. Does integration give an Accurate or Approximate answer?

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What do you mean by "reality"? – Gerry Myerson Sep 24 '12 at 1:26
I mean is it really possible to divide a figure into infinite regions? Because as we keep on dividing an irregular shape, always there is some more region waiting to be divided into infinite regions. – Ranjan Yajurvedi Sep 24 '12 at 1:31
What do you mean by "really possible"? Is this a question about Mathematics, or about human physiology? – Gerry Myerson Sep 24 '12 at 1:44
In reality, it is not possible to create a perfect cube. A mathematical cube has volume exactly the cube of the side, but a physical one does not. Similarly for a sphere-the mathematical one has volume exactly $\frac 42 \pi r^3$ but the physical one does not. – Ross Millikan Sep 24 '12 at 3:49

The act of division is one method to calculate an integration value, it is a numerical approach. If you use such a method, accuracy of result will depend on the function and number of intervals involved. However, if you don't use the numerical approach and use integration by finding a function, then the value is as accurate as you can calculate f(b) and f(a) - To illustrate, if you know $f$ for a given $F$ such that:

$$\int_{a}^{b} F(x) dx =f(b)-f(a)$$

the result would be as accurate as the calculation of f(b)-f(a) can be computed.

Examples:

Find the area under the curve of the function $F(x) = ln(x)$ between $1$ and $3$ using a numerical method and using a function.

Using a numerical method:

If you use 5 intervals, then we get $Area = 1.2870135405$

If you use 50 intervals, then we get $Area = 1.295748$

Using a function:

Given $F(x)=ln(x)$ then $f(x)=xln(x)-x+c$

The desired Area is

$$\int_{1}^{3} ln(x) dx =(3*ln(3)-3)-(1*ln(1)-1)=1.295836866$$

This last result is more accurate than the previous ones. However, it is as accurate as we could calculate the value of the $ln$.

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I'm sorry, but i don't get you – Ranjan Yajurvedi Sep 24 '12 at 1:33
Which part is not clear? – NoChance Sep 24 '12 at 1:33
Integration by finding a function – Ranjan Yajurvedi Sep 24 '12 at 1:40
OK, I will edit the answer to be more clear. – NoChance Sep 24 '12 at 1:46
It is easy to find such a function! It can be (for example) $\int_a^x F(t)dt$. :) – tomasz Sep 24 '12 at 1:48