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?
Integration - Accurate or Approximate
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0In 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. – 2012-09-24
1 Answers
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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0Thanks for your feedback, glad to be of help. – 2012-09-24