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

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    2016
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The integral challenge from POTW #231 involves evaluating the expression ∫_{0}^{1} (√[4]{1-x^7} - √[7]{1-x^4}) dx. Participants are encouraged to follow the guidelines provided for submitting solutions. The discussion highlights Theia's successful solution to the problem. A model solution is also available for reference. This integral presents an interesting challenge in calculus, combining roots and polynomial expressions.
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Here is this week's POTW:

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

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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!
 
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Congratulations to Theia for his correct solution::)

Here's the model solution:

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.

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

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