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H-x-8-x-2-and-f-x-x-2-3x-6-2x-4-1-Calculate-the-surface-V-n-of-area-limited-by-the-the-line-x-6-x-6-n-n-N-and-the-curve-of-H-x-and-f-x-in-function-of-n-2-Knowing-that-1-2-2-2-




Question Number 137321 by mathocean1 last updated on 01/Apr/21
H(x)=(8/(x−2)) and f(x)=((x^2 +3x+6)/(2x−4))  1)Calculate the surface V_n  of  area limited by the the line  x=6; x=6+n (n∈N^∗ ) and the  curve of H(x) and f(x) in function  of n.  2) Knowing that 1^2 +2^2 +...+n^2 =((n(n+1)(2n+1))/6)  Deduct the value of   S_n =V_1 +V_2 +...+V_(n ) in funtion  of n .  3) Determinate the smallest  integer n such that S_n >100
$${H}\left({x}\right)=\frac{\mathrm{8}}{{x}−\mathrm{2}}\:{and}\:{f}\left({x}\right)=\frac{{x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{6}}{\mathrm{2}{x}−\mathrm{4}} \\ $$$$\left.\mathrm{1}\right){Calculate}\:{the}\:{surface}\:{V}_{\boldsymbol{{n}}} \:{of} \\ $$$${area}\:{limited}\:{by}\:{the}\:{the}\:{line} \\ $$$${x}=\mathrm{6};\:{x}=\mathrm{6}+\boldsymbol{{n}}\:\left({n}\in\mathbb{N}^{\ast} \right)\:{and}\:{the} \\ $$$${curve}\:{of}\:{H}\left({x}\right)\:{and}\:{f}\left({x}\right)\:{in}\:{function} \\ $$$${of}\:\boldsymbol{{n}}. \\ $$$$\left.\mathrm{2}\right)\:{Knowing}\:{that}\:\mathrm{1}^{\mathrm{2}} +\mathrm{2}^{\mathrm{2}} +…+{n}^{\mathrm{2}} =\frac{{n}\left({n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{1}\right)}{\mathrm{6}} \\ $$$${Deduct}\:{the}\:{value}\:{of}\: \\ $$$${S}_{{n}} ={V}_{\mathrm{1}} +{V}_{\mathrm{2}} +…+{V}_{{n}\:} {in}\:{funtion} \\ $$$${of}\:{n}\:. \\ $$$$\left.\mathrm{3}\right)\:{Determinate}\:{the}\:{smallest} \\ $$$${integer}\:{n}\:{such}\:{that}\:{S}_{{n}} >\mathrm{100} \\ $$

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