Lets-say-we-have-three-points-A-0-0-B-x-y-C-x-y-Assuming-that-both-B-and-C-are-point-on-a-fuction-y-f-x-we-can-calculate-the-area-under-the-point-where-it-makes-a-right-triangle-with-the-o

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Question Number 4535 by FilupSmith last updated on 05/Feb/16

$$\mathrm{Lets}\mathrm{say}\mathrm{we}\mathrm{have}\mathrm{three}\mathrm{points}:$$$$A(0,0)$$$$B(x,y)$$$$C(\delta x,\delta y)$$$$$$$$\mathrm{Assuming}\mathrm{that}\mathrm{both}B\mathrm{and}C\mathrm{are}\mathrm{point}$$$$\mathrm{on}\mathrm{a}\mathrm{fuction}y=f\left(x\right),\mathrm{we}\mathrm{can}\mathrm{calculate}$$$$\mathrm{the}\mathrm{area}\mathrm{under}\mathrm{the}\mathrm{point}\mathrm{where}\mathrm{it}\mathrm{makes}$$$$\mathrm{a}\mathrm{right}\mathrm{triangle}\mathrm{with}\mathrm{the}\mathrm{origin}\mathrm{and}\mathrm{x}-\mathrm{axis}.$$$$$$$$\mathrm{Can}\mathrm{we}\mathrm{calculate}\mathrm{the}\mathrm{area}\mathrm{of}ABC?$$

Commented by FilupSmith last updated on 05/Feb/16

Commented by Yozzii last updated on 07/Feb/16

$$tan\alpha =y/x,tan\varphi =\delta y/\delta x$$$$\Rightarrow \alpha -\varphi ={tan}^{-1}\left(y/x\right)-{tan}^{-1}\left(\delta y/\delta x\right)$$$$A_{△ABC}=\frac{1}{2}\sqrt{(y^2+x^2)({\left\{\delta y\right\}}^2+{\left\{\delta x\right\}}^2)}\left[sin\{{tan}^{-1}\frac{y}{x}-{tan}^{-1}\frac{\delta y}{\delta x}\}\right]$$

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