Question and Answers Forum

AllQuestion and Answers: Page 1786

Question Number 31474 Answers: 1 Comments: 1

A vacuum chamber has a door with surface dimensions 20cm×20cm.How many 70kg men, each exerting a pull equal to his weight,will it take to pull the door open when the chamber is evacuated to a pressure of 0.1 atm?

$${A}\:{vacuum}\:{chamber}\:{has}\:{a}\:{door} \\ $$$${with}\:{surface}\:{dimensions} \\ $$$$\mathrm{20}{cm}×\mathrm{20}{cm}.{How}\:{many}\:\mathrm{70}{kg}\:{men}, \\ $$$${each}\:{exerting}\:{a}\:{pull}\:{equal}\:{to}\:{his} \\ $$$${weight},{will}\:{it}\:{take}\:{to}\:{pull}\:{the} \\ $$$${door}\:{open}\:{when}\:{the}\:{chamber}\:{is} \\ $$$${evacuated}\:{to}\:{a}\:{pressure}\:{of}\:\mathrm{0}.\mathrm{1}\:{atm}? \\ $$

Question Number 31466 Answers: 0 Comments: 0

let give I_n = ∫_(1/n) ^1 (√(1+t^2 )) dt 1) calculate I_n 2) find lim_(n→∞) I_n .

$${let}\:{give}\:{I}_{{n}} =\:\int_{\frac{\mathrm{1}}{{n}}} ^{\mathrm{1}} \:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{I}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow\infty} \:{I}_{{n}} \:\:\:. \\ $$

Question Number 31465 Answers: 0 Comments: 0

find F(α)= ∫_0 ^1 ((arctan(αx))/(1+x^2 )) dx with α ∈ R−{1,−1}

$${find}\:\:{F}\left(\alpha\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{arctan}\left(\alpha{x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:\:{with}\:\alpha\:\in\:{R}−\left\{\mathrm{1},−\mathrm{1}\right\} \\ $$

Question Number 31464 Answers: 0 Comments: 0

1) find A_n = ∫_0 ^(π/2) e^(−x) cos(nx)dx 2) find S_n = Σ_(k=0) ^n A_k .

$$\left.\mathrm{1}\right)\:{find}\:\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:{e}^{−{x}} {cos}\left({nx}\right){dx} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:{S}_{{n}} =\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{A}_{{k}} \:\:. \\ $$

Question Number 31463 Answers: 0 Comments: 0

let give the function f(x)=∫_0 ^π ln(1+xcosθ)dθ with ∣x∣<1 1) find a simple form of f(x) 2)calculate ∫_0 ^π ln(1−cosθ)dθ 3)calculate ∫_0 ^π ln(1+cosθ)dθ.

$${let}\:{give}\:{the}\:{function}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\pi} {ln}\left(\mathrm{1}+{xcos}\theta\right){d}\theta\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\pi} \:{ln}\left(\mathrm{1}−{cos}\theta\right){d}\theta \\ $$$$\left.\mathrm{3}\right){calculate}\:\int_{\mathrm{0}} ^{\pi} {ln}\left(\mathrm{1}+{cos}\theta\right){d}\theta. \\ $$

Question Number 31462 Answers: 0 Comments: 0

find ∫_0 ^∞ ((arctan(x+(1/x)))/(1+x^2 ))dx.

$${find}\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{{arctan}\left({x}+\frac{\mathrm{1}}{{x}}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx}. \\ $$

Question Number 31461 Answers: 0 Comments: 0

let give u_n = Σ_(k=1) ^n sin ((k/n)) sin((k/n^2 )) 1) prove that the sequence (u_n ) is convergent 2) find lim_(n→∞) u_n .

$${let}\:{give}\:\:{u}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{sin}\:\left(\frac{{k}}{{n}}\right)\:{sin}\left(\frac{{k}}{{n}^{\mathrm{2}} }\right) \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{the}\:{sequence}\:\left({u}_{{n}} \right)\:{is}\:{convergent} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow\infty} \:{u}_{{n}} . \\ $$

Question Number 31460 Answers: 0 Comments: 1

find in terms of n the value of A_n = ∫_0 ^1 Π_(k=1) ^(n−1) (x^2 −2xcos(((kπ)/n)) +1)dx with n from N^★ .

$${find}\:{in}\:{terms}\:{of}\:\:{n}\:{the}\:{value}\:{of} \\ $$$${A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\prod_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \left({x}^{\mathrm{2}} \:−\mathrm{2}{xcos}\left(\frac{{k}\pi}{{n}}\right)\:+\mathrm{1}\right){dx}\:\:\:{with}\:{n}\:{from}\:{N}^{\bigstar} . \\ $$

Question Number 31456 Answers: 0 Comments: 2

Find sum of S= (2/3) + (4/3^2 ) + (6/3^3 ) + (8/3^4 ) +......+∞ ?