Sharp EL-W506T vs. Sharp EL-W516T

[Mjim]

I tried this myself, but I kept getting different results. I downloaded Xcas to my PC (https://www-fourier.ujf-grenoble.fr/~parisse/giac.html I think the same engine behind the HP Prime):

Using: G=6.67430*10^-11, rE = 6.371*10^6, mE = 5.972*10^24, 1ly (in metres) = 299,792,458*3600*24*365.25
C = G*mE*1000 = 3.98589*10^17

E = integrate(C/r^2, rE, 1ly):
Wolfram (only gave 6 significant figures): 6.25631*10^10
Xcas: 6.25630506563*10^10
Casio fx-9750GII: 6.256305066*10^10
Casio fx-570MS (upgraded temporarily from a fx-82MS): Throws math error.
Sharp EL-W516X:
-Default divisions: 1.54840203*10^17
-1024 divisions: 6.694153651*10^10
-4096 divisions: 6.258289041*10^10
-32768 divisions: 6.251558833*10^10 (This took a very...very long time)

Not sure if these results are right; might of made a mistake with some units somewhere.

The Sharp is way off, part of it could be internal digit accuracy. If you move the constant C outside of the integral the Casio fx-9750GII will give 29390.80513 as an answer which will actually increase if we decrease the distance to 1 light second where it pops back up to 6.1233*10^10. The Xcas result is perfect either way, so perhaps the Prime will be similar?

The Casio fx-82MS error algorithm basically told us to go away, which is a better outcome since there is no way it could give a good approximation using the Simpson algorithm, if even the Sharp at 1024 divisions (512 more then the max on the Casio MS), couldn't even get a single significant figure down.

Good example for showing the weakness in the Simpson Integration algorithm. I have to agree that I don't trust the Sharp for integration, but to be fair, any calculator using the Simpson algorithm will likely have the same issue; the Casio 570MS just clearly recognizes when it won't be able to deliver a good result and throws a math error, but any other calculator with the same Simpson Algorithm will probably suffer similarly.

BTW, with this integration example, 1/r^p is convergent if p > 1; so since p = 2 we know that this function is convergent. The energy needed to offset earth's gravitational influence is essentially a fixed value beyond a certain distance. In any case, this means there isn't much difference between 1 light day and 1 light year in terms of energy required, which makes sense as earths gravity is pretty much negligible after traveling 173 AU or 173 times the distance from earth to the sun).