Quiz: calculating a definite integral

12302013, 09:37 PM
(This post was last modified: 01022014 05:43 PM by Thomas Klemm.)
Post: #1




Quiz: calculating a definite integral
Let's assume we want to calculate the following definite integral:
\[ \int_{0}^{1}x^{x}dx \] We'd use a calculator with an \( \int_{x}^{y} \) button and write a short program: 01 LBL 0 02 CHS 03 \(y^x\) 04 RTN The result is something like: 1.291285997 Now what if for some reason this button doesn't work or we don't have a calculator that supports numerical integration? How can we still calculate this definite integral? Cheers Thomas Addendum: How can I guide you into the right direction without spoiling the quiz? Of course you can just use another calculator where the button isn't broken. But that's not the point of this quiz. 

12302013, 10:46 PM
Post: #2




RE: Quiz: calculating a defined integral
I would use Excel and evaluate using the trapezoidal rule.
Regards, John 

12302013, 11:44 PM
(This post was last modified: 12302013 11:46 PM by W_Max.)
Post: #3




RE: Quiz: calculating a defined integral
Well... It depends on what machine we have. Even plain TI55 can calculate this, using primitive rectangle method. But I prefer Simpson's if it can fit in program memory, of course


12312013, 04:24 AM
(This post was last modified: 12312013 04:29 AM by Namir.)
Post: #4




RE: Quiz: calculating a defined integral
I do believe the program should be:
Code: 01 LBL 0 For the builtin integral to work! your answer is correct! Now to answer your question. I would use an open NewtonCote numerical integration method since this class of methods do not evaluate the integral at the end points, just in case my machine barks at evaluating 1/0^0. Namir 

12312013, 04:56 AM
(This post was last modified: 12312013 08:58 AM by Thomas Klemm.)
Post: #5




RE: Quiz: calculating a defined integral
(12312013 04:24 AM)Namir Wrote: I do believe the program should be:The HP15C conveniently fills the stack with \(x\) when calling your function. Quote:I would use an open NewtonCote numerical integration method since this class of methods do not evaluate the integral at the end points, just in case my machine barks at evaluating 1/0^0. Good point. The function is continuous even at 0: \(\lim_{x\to0}x^{x}=1\) But the first derivative is singular at 0: \(\lim_{x\to0}\frac{\partial x^{x}}{\partial x}=\infty\) HP's modified Romberg method avoids the endpoints of the interval as well. 

12312013, 06:37 AM
Post: #6




RE: Quiz: calculating a defined integral
(12302013 11:44 PM)W_Max Wrote: Well... It depends on what machine we have. Even plain TI55 can calculate this, using primitive rectangle method. But I prefer Simpson's if it can fit in program memory, of courseI've tried Valentin's Gaussian integration for the HP11C for various values of N: 10: 1.291303564 20: 1.291290482 50: 1.291286729 100: 1.291286183 200: 1.291286042 500: 1.291286006 1000: 1.291285997 He doesn't seem to be fond of Simpson's rule: Quote:Simpson's method is to numerical integration what bubble sort is to sorting, i.e., vastly inefficient. 

12312013, 07:13 AM
Post: #7




RE: Quiz: calculating a defined integral
Interesting is the suggestion that a Gaussian method would have been used in the 34C if there were ROM space. I originally used such a method in the 34S and almost everyone wanted it changed to an adaptive one instead (Romberg).
 Pauli 

12312013, 08:54 AM
Post: #8




RE: Quiz: calculating a defined integral  
12312013, 09:14 AM
Post: #9




RE: Quiz: calculating a defined integral  
12312013, 10:12 AM
Post: #10




RE: Quiz: calculating a defined integral  
12312013, 10:24 AM
(This post was last modified: 12312013 10:51 AM by peacecalc.)
Post: #11




RE: Quiz: calculating a defined integral
Hello Thomas,
If I use fix 4 on the hp 15c (the very old version no LE) I get in some seconds a correct result (like N=20, some posts above), that shows the algorithem functions well. Maybe the problems by fix 9 are caused by calculating the integrand: instead of \[ x^{x} \] as \[ \exp(x\cdot\ln(x)) \]. That have a great alteration rate near zero. The hp 15c algorithem don't calculate at the limits the value of the integrand, but for getting a higher precision it runs into a great number of calculations. Sincerely peaceglue P.S. I have to correct myself, even with 0.01 or 0.1 it doesn't come to an end. 

12312013, 10:57 AM
Post: #12




RE: Quiz: calculating a defined integral
Hello all,
now the result for the limits 0,1 and 1 take a few minutes: 1,176021571 (fix 9). But that's not a resonable duration. In fix 4 it takes some seconds. Sincerely peacecalc 

12312013, 01:14 PM
Post: #13




RE: Quiz: calculating a defined integral
It takes 2'27" to calculate this integral on a DM15CC with FIX 9.
It takes 28" to do it with the RPN15C emulator on my iPhone. On a real HP15C it takes probably much longer. Can you do it much faster? Cheers Thomas 

12312013, 01:29 PM
Post: #14




RE: Quiz: calculating a defined integral
Plot the function, print out the plot and use your trusty planimeter to calculate the area. If you are unlucky enough to not own a planimeter, hatchet planimeters are easy to construct.
 Pauli 

12312013, 02:16 PM
Post: #15




RE: Quiz: calculating a defined integral
Plot the function, cut out the plot and use your trusty weighing scale to calculate the area. A chemist's scale shall do.
d:) 

12312013, 02:20 PM
Post: #16




RE: Quiz: calculating a defined integral
Hello Thomas,
it is allowed to calculate without a calc with an approximation? But then I never can beat 28'', therefore I'm to slow... 

12312013, 05:55 PM
Post: #17




RE: Quiz: calculating a defined integral
(12312013 02:20 PM)peacecalc Wrote: Hello Thomas,Sure! Use whatever you like. I used paper and pencil first. And only then the calculator and was down at 1". Best regards Thomas 

12312013, 06:01 PM
Post: #18




RE: Quiz: calculating a defined integral  
01012014, 10:31 PM
Post: #19




RE: Quiz: calculating a definite integral  
01022014, 08:11 AM
Post: #20




RE: Quiz: calculating a definite integral  
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