More Simpson Rule Question

Dec 2008
509
2
Hi

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

\(\displaystyle x_0 = 0\)

\(\displaystyle x_1 = \frac{1}{4}\)

\(\displaystyle x_2 = \frac{2}{4}\)

\(\displaystyle x_3 = \frac{3}{4}\)

\(\displaystyle x_4 = \frac{4}{4}\)

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

=0.70553

book's answer is 0.7889
 

CaptainBlack

MHF Hall of Fame
Nov 2005
14,972
5,271
someplace
Hi

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

\(\displaystyle x_0 = 0\)

\(\displaystyle x_1 = \frac{1}{4}\)

\(\displaystyle x_2 = \frac{2}{4}\)

\(\displaystyle x_3 = \frac{3}{4}\)

\(\displaystyle x_4 = \frac{4}{4}\)

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

=0.70553

book's answer is 0.7889
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
 
Dec 2008
509
2
Hi

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

\(\displaystyle x_0 = 0\)

\(\displaystyle x_1 = \frac{1}{4}\)

\(\displaystyle x_2 = \frac{2}{4}\)

\(\displaystyle x_3 = \frac{3}{4}\)

\(\displaystyle x_4 = \frac{4}{4}\)

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

=0.70553

book's answer is 0.7889
srry about that, the above is the correct equation.
 

CaptainBlack

MHF Hall of Fame
Nov 2005
14,972
5,271
someplace
\(\displaystyle \int_0^1 x^{x} dx = \frac{1}{12}[1+4(\frac{1}{4}^{\frac{1}{4}})+2(\frac{1}{2}^{\frac{1}{2}})+4(\frac{3}{4}^{\frac{3}{4}})+1]\)

=0.7889
 
Dec 2008
509
2
\(\displaystyle \int_0^1 x^{x} dx = \frac{1}{12}[1+4(\frac{1}{4}^{\frac{1}{4}})+2(\frac{1}{2}^{\frac{1}{2}})+4(\frac{3}{4}^{\frac{3}{4}})+1]\)

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

CaptainBlack

MHF Hall of Fame
Nov 2005
14,972
5,271
someplace
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