# Thread: More Simpson Rule Question

1. ## More Simpson Rule Question

Hi

The following question i don't know why it is incorrect.
1)$\displaystyle \int_1^3 x^{x}dx$ n=4

$\displaystyle x_0 = 0$

$\displaystyle x_1 = \frac{1}{4}$

$\displaystyle x_2 = \frac{2}{4}$

$\displaystyle x_3 = \frac{3}{4}$

$\displaystyle x_4 = \frac{4}{4}$

$\displaystyle \int_0^1 x^{x} dx = \frac{1}{12}[0+4(\frac{1}{4}^{\frac{1}{4}})+2(\frac{1}{2}^{\fra c{1}{2}})+4(\frac{3}{4}^{\frac{3}{4}})+1]$

=0.70553

2. Originally Posted by Paymemoney
Hi

The following question i don't know why it is incorrect.
1)$\displaystyle \int_1^3 x^{x}dx$ n=4

$\displaystyle x_0 = 0$

$\displaystyle x_1 = \frac{1}{4}$

$\displaystyle x_2 = \frac{2}{4}$

$\displaystyle x_3 = \frac{3}{4}$

$\displaystyle x_4 = \frac{4}{4}$

$\displaystyle \int_0^1 x^{x} dx = \frac{1}{12}[0+4(\frac{1}{4}^{\frac{1}{4}})+2(\frac{1}{2}^{\fra c{1}{2}})+4(\frac{3}{4}^{\frac{3}{4}})+1]$

=0.70553

Why have you used $\displaystyle 0^0=0$ in this case? You need $\displaystyle \lim_{x \to 0}x^x=1$

Next time you post a question try to make it clear what your question is, and don't change the detail part way through.

CB

3. Originally Posted by Paymemoney
Hi

The following question i don't know why it is incorrect.
1)$\displaystyle \int_0^1 x^{x}dx$ n=4

$\displaystyle x_0 = 0$

$\displaystyle x_1 = \frac{1}{4}$

$\displaystyle x_2 = \frac{2}{4}$

$\displaystyle x_3 = \frac{3}{4}$

$\displaystyle x_4 = \frac{4}{4}$

$\displaystyle \int_0^1 x^{x} dx = \frac{1}{12}[0+4(\frac{1}{4}^{\frac{1}{4}})+2(\frac{1}{2}^{\fra c{1}{2}})+4(\frac{3}{4}^{\frac{3}{4}})+1]$

=0.70553

srry about that, the above is the correct equation.

4. $\displaystyle \int_0^1 x^{x} dx = \frac{1}{12}[1+4(\frac{1}{4}^{\frac{1}{4}})+2(\frac{1}{2}^{\fra c{1}{2}})+4(\frac{3}{4}^{\frac{3}{4}})+1]$

=0.7889

5. Originally Posted by CaptainBlack
$\displaystyle \int_0^1 x^{x} dx = \frac{1}{12}[1+4(\frac{1}{4}^{\frac{1}{4}})+2(\frac{1}{2}^{\fra c{1}{2}})+4(\frac{3}{4}^{\frac{3}{4}})+1]$

=0.7889
but isn't $\displaystyle x_0$ equal to 0?

6. Originally Posted by Paymemoney
but isn't $\displaystyle x_0$ equal to 0?
No that's why I said it's not in post #2 in this thread. $\displaystyle 0^0$ is undefined here you need to use $\displaystyle \lim_{x\to 0}x^x=1$ for the value of the integrand at $\displaystyle x=0$

CB