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enter image description here

This demonstrates that $\int_0^1 t^{p/q} + t^{q/p} dt = 1$. Could you please explain how the proof without words shows that?

Brian M. Scott
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user103493
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  • I opened this thinking you meant "Without words, Explain this calculus proof." and I hoped someone answered it with only a picture. – Paul Totzke Jun 22 '19 at 20:37

2 Answers2

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The functions $t^{p/q}$ and $t^{q/p}$ are inverses of each other, since

$$\left(t^{p/q}\right)^{q/p} = t$$

and vice-versa.

So the graph of one can be found by flipping the graph of the other over the line $y = x$. In particular, the lightly shaded area is the exact same as if we were to just graph $t^{q/p}$ and shade the area under that graph.

2

$\int_0^1 t^{p/q} dt = \int_0^1 y^{p/q} dy = $ the light area in the graph; $\int_0^1 t^{q/p} dt = \int_0^1 x^{q/p} dx $ = the dark area in the graph.

Magdiragdag
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