Solve with integrating an implicit function

[Alberto Candel]

(06-20-2014 02:40 PM)Helge Gabert Wrote:  I recently came, by chance, across this interesting post from one of the hpmuseum archives, originally posted by Valentin Albillo

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Test 7 - Solving a definite integral of an implicit function:
Find X in [0,1] such that

/X
|
| y(x).dx = 1/3
|
/0
where y(x) is an ultra-radical function (a member of the family of elliptic functions) implicitly defined by:
5
y + y = x
using precisions of 1E-3 and 1E-6 for the integral.

Helge,
I apologize but I could not resist "hijacking" your post. It turns out that I am currently teaching Adv Calc.
and your question lead me to some good assignment for my students (some have the Prime). One problem (from Spivak's) was to show that there is a function f(x) such that (f(x))^5+(f(x))=x. Your question then leads to applications of the Fund Thm of Calc and to the Intermediate Value Thm. So, thanks!

BTW, it is really interesting to note how fast the Advanced Graphics of the HP Prime is for plotting the implicit function y^5+y=x compared to plotting F1(x):=solve(y^5+y=x,y) in the Function App