Question Number 137321 by mathocean1 last updated on 01/Apr/21
$${H}\left({x}\right)=\frac{\mathrm{8}}{{x}−\mathrm{2}}\:{and}\:{f}\left({x}\right)=\frac{{x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{6}}{\mathrm{2}{x}−\mathrm{4}} \\ $$$$\left.\mathrm{1}\right){Calculate}\:{the}\:{surface}\:{V}_{\boldsymbol{{n}}} \:{of} \\ $$$${area}\:{limited}\:{by}\:{the}\:{the}\:{line} \\ $$$${x}=\mathrm{6};\:{x}=\mathrm{6}+\boldsymbol{{n}}\:\left({n}\in\mathbb{N}^{\ast} \right)\:{and}\:{the} \\ $$$${curve}\:{of}\:{H}\left({x}\right)\:{and}\:{f}\left({x}\right)\:{in}\:{function} \\ $$$${of}\:\boldsymbol{{n}}. \\ $$$$\left.\mathrm{2}\right)\:{Knowing}\:{that}\:\mathrm{1}^{\mathrm{2}} +\mathrm{2}^{\mathrm{2}} +...+{n}^{\mathrm{2}} =\frac{{n}\left({n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{1}\right)}{\mathrm{6}} \\ $$$${Deduct}\:{the}\:{value}\:{of}\: \\ $$$${S}_{{n}} ={V}_{\mathrm{1}} +{V}_{\mathrm{2}} +...+{V}_{{n}\:} {in}\:{funtion} \\ $$$${of}\:{n}\:. \\ $$$$\left.\mathrm{3}\right)\:{Determinate}\:{the}\:{smallest} \\ $$$${integer}\:{n}\:{such}\:{that}\:{S}_{{n}} >\mathrm{100} \\ $$