- #1

- 2

- 0

- Thread starter shreyarora
- Start date

- #1

- 2

- 0

- #2

- 22,089

- 3,292

Can you please expand your question a bit?

What do you mean with "normal definite integration", to my knowledge that simply is the Riemann integral?

And what do you mean with accuracy? Riemann integration is 100% accurate, it uses no approximations what-so-ever.

We'll be in a better position to answer you if you let is know where you're coming from...

- #3

HallsofIvy

Science Advisor

Homework Helper

- 41,833

- 961

Shreyarora, the formula for the accuracy is given in any Calculus book. For example, on page 487 of Salas, Hille, and Etgen's

[tex]\int_a^b f(t)dt[/tex]

is given as less than

[tex](f(b)- f(a))\frac{b- a}{n}[/tex]

So, for example, if you use 10 rectangles to integrate

[tex]\int_0^1 x^2 dx[/tex]

Your error would be less than (1-0)((1- 0)/10) or 1/10.

Of course, the trapezoidal method and Simpson's rule give better accuracy.

- #4

- 2

- 0

By normal integration I meant that, if you integrate x^2 from a to b, you substitute limits to x^3/3.

whereas, the computation differs while evaluating integral using Riemann Integration.

I am actually writing an article wherein I have to justify that using Riemann Integration yields accurate results to a real life problem over the "normal definite integration" that I have defined above.

Or is it actually possible to compare the two methods?

- #5

- 306

- 1

[tex]

\int_1^2 x^2 dx=\frac{1}{3}x^3|_1^2=\frac{7}{3}

[/tex]

I just performed Riemann integration. But I could I have

[tex]

\sum_{k=0}^{N-1} (1+k\Delta x)^2 \Delta x

[/tex]

where [tex]\Delta x =(2-1)/N[/tex]

- Last Post

- Replies
- 13

- Views
- 3K

- Last Post

- Replies
- 19

- Views
- 7K

- Last Post

- Replies
- 8

- Views
- 3K

- Last Post

- Replies
- 28

- Views
- 2K

- Replies
- 1

- Views
- 2K

- Last Post

- Replies
- 5

- Views
- 2K

- Last Post

- Replies
- 2

- Views
- 2K

- Replies
- 30

- Views
- 8K

- Last Post

- Replies
- 3

- Views
- 5K

- Last Post

- Replies
- 1

- Views
- 2K