How Do You Solve the Integral Challenge from POTW #231?

[anemone]

Here is this week's POTW:

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Evaluate \(\displaystyle \int_{0}^{1}\left(\sqrt[4]{1-x^7}-\sqrt[7]{1-x^4}\right) dx\).

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[anemone]

Congratulations to Theia for his correct solution::)

Here's the model solution:

Spoiler

Note that for $x,\,y\ge 0$, it holds that $y=\sqrt[7]{1-x^4}$ is equivalent to $y^7+x^4=1$, or $x=\sqrt[4]{1-y^7}$, therefore the two functions are inverse to each other and since both map $[0,\,1]$ to $[0,\,1]$ the two areas must be the same, i.e.

\(\displaystyle \int_{0}^{1}\left(\sqrt[4]{1-x^7}\right) dx=\int_{0}^{1}\left(\sqrt[7]{1-x^4}\right) dx\)

\(\displaystyle \therefore \int_{0}^{1}\left(\sqrt[4]{1-x^7}-\sqrt[7]{1-x^4}\right) dx=0\)