Golden ratio

Calculus Level 4

$\large \displaystyle\int _{ 0 }^{ 2 }{ \left\lfloor { x }^{ 2 }-x \right\rfloor \, dx } =a-b\phi$

The equation above holds true for positive integers $a$ and $b$ , where $\phi = \frac{1+\sqrt5}2$ is the golden ratio .

Find $a+b$ .

The answer is 2.

1 solution

Rishabh Jain
Apr 6, 2016

We know: $-1<x^2-x<0 \text{ when }~\color{#D61F06}{ 0<x<1}~~\\ 0<x^2-x<1\text{ when } ~\color{#D61F06}{1<x<\phi}~~\\ 1<x^2-x<2 \text{ when } ~\color{#D61F06}{\phi<x<2}~~$

The integration is therefore:

$\int_0^1(-1)\mathrm{d}x +\int_{1}^{\phi}(0)\mathrm{d}x+\int_{\phi}^2(1)dx$ $=1-\phi$

$\huge\therefore~ 1+1=\boxed{2}$

Golden ratio

The answer is 2.

1 solution

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