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IntegrationQuestion and Answers: Page 125

Question Number 124976 Answers: 2 Comments: 1

(1) The gravitational force (in lb) of attraction between two objects is given by F =(k/x^2 ), where x is the distance between the objects. If the objects are 10 ft apart, find the work required to separate them until they are 50 ft apart. Express the result in terms of k. (a) (k/(500)) (b) ((2k)/(25)) (c) (k/5) (d) (k/(40)) (2)One end of a pool is vertical wall 15 ft wide. What is the force exerted on this wall by the water if it is 6 ft deep? The density of water is 62.4 lb/ft^3 (a) 8420 lb (b) 33,700 lb (c) 2810 lb (d) 16,800 lb (3)Find the area of the surface generated by revolving the curve about that indicated axis. x = 3(√(4−y)) , 0≤y≤((15)/4) , y−axis (a) (((125)/2)+5(√(10)))π (b) (((125)/2)−5(√(10)))π (c) ((125)/2)π (d) 5π(√(10))

$$\left(\mathrm{1}\right)\:{The}\:{gravitational}\:{force}\:\left({in}\:{lb}\right)\:{of} \\ $$$${attraction}\:{between}\:{two}\:{objects}\:{is}\:{given} \\ $$$${by}\:{F}\:=\frac{{k}}{{x}^{\mathrm{2}} },\:{where}\:{x}\:{is}\:{the}\:{distance} \\ $$$${between}\:{the}\:{objects}.\:{If}\:{the}\:{objects}\:{are} \\ $$$$\mathrm{10}\:{ft}\:{apart},\:{find}\:{the}\:{work}\:{required}\:{to} \\ $$$${separate}\:{them}\:{until}\:{they}\:{are}\:\mathrm{50}\:{ft}\:{apart}.\:{Express} \\ $$$${the}\:{result}\:{in}\:{terms}\:{of}\:{k}. \\ $$$$\left({a}\right)\:\frac{{k}}{\mathrm{500}}\:\:\:\:\:\:\left({b}\right)\:\frac{\mathrm{2}{k}}{\mathrm{25}}\:\:\:\:\:\left({c}\right)\:\frac{{k}}{\mathrm{5}}\:\:\:\left({d}\right)\:\frac{{k}}{\mathrm{40}} \\ $$$$\left(\mathrm{2}\right){One}\:{end}\:{of}\:{a}\:{pool}\:{is}\:{vertical}\:{wall}\:\mathrm{15}\:{ft} \\ $$$${wide}.\:{What}\:{is}\:{the}\:{force}\:{exerted}\:{on}\:{this} \\ $$$${wall}\:{by}\:{the}\:{water}\:{if}\:{it}\:{is}\:\mathrm{6}\:{ft}\:{deep}? \\ $$$${The}\:{density}\:{of}\:{water}\:{is}\:\mathrm{62}.\mathrm{4}\:{lb}/{ft}^{\mathrm{3}} \\ $$$$\left({a}\right)\:\mathrm{8420}\:{lb}\:\:\:\:\left({b}\right)\:\mathrm{33},\mathrm{700}\:{lb}\:\:\:\:\left({c}\right)\:\mathrm{2810}\:{lb}\:\:\left({d}\right)\:\mathrm{16},\mathrm{800}\:{lb} \\ $$$$\left(\mathrm{3}\right){Find}\:{the}\:{area}\:{of}\:{the}\:{surface}\:{generated} \\ $$$${by}\:{revolving}\:{the}\:{curve}\:{about}\:{that}\: \\ $$$${indicated}\:{axis}.\:\:{x}\:=\:\mathrm{3}\sqrt{\mathrm{4}−{y}}\:,\:\mathrm{0}\leqslant{y}\leqslant\frac{\mathrm{15}}{\mathrm{4}}\:,\:{y}−{axis} \\ $$$$\left({a}\right)\:\left(\frac{\mathrm{125}}{\mathrm{2}}+\mathrm{5}\sqrt{\mathrm{10}}\right)\pi\:\:\:\:\:\:\:\left({b}\right)\:\left(\frac{\mathrm{125}}{\mathrm{2}}−\mathrm{5}\sqrt{\mathrm{10}}\right)\pi \\ $$$$\left({c}\right)\:\frac{\mathrm{125}}{\mathrm{2}}\pi\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({d}\right)\:\mathrm{5}\pi\sqrt{\mathrm{10}}\: \\ $$$$ \\ $$

Question Number 124957 Answers: 1 Comments: 1

Question Number 124922 Answers: 0 Comments: 0

let f(x)=((ln(1+2x))/(x^2 +1)) 1) calculste f^((n)) (x) and f^((n)) (0) 2)develop f at integr serie 3) find ∫_0 ^1 f(x)dx

$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{2x}\right)}{\mathrm{x}^{\mathrm{2}} +\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:\mathrm{calculste}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{x}\right)\:\mathrm{and}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\mathrm{develop}\:\mathrm{f}\:\mathrm{at}\:\mathrm{integr}\:\mathrm{serie} \\ $$$$\left.\mathrm{3}\right)\:\mathrm{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \\ $$

Question Number 124921 Answers: 0 Comments: 0

calculate ∫_(−∞) ^(+∞) z^(−x^2 ) dx with z complex

$$\mathrm{calculate}\:\int_{−\infty} ^{+\infty} \:\mathrm{z}^{−\mathrm{x}^{\mathrm{2}} } \mathrm{dx}\:\:\mathrm{with}\:\mathrm{z}\:\mathrm{complex} \\ $$

Question Number 124920 Answers: 2 Comments: 0

calculate ∫_0 ^∞ e^(−x^n ) dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\mathrm{e}^{−\mathrm{x}^{\mathrm{n}} } \mathrm{dx}\: \\ $$

Question Number 124919 Answers: 0 Comments: 2

find U_n =∫_0 ^1 x^n arctan(x)dx with n integr nstural

$$\mathrm{find}\:\mathrm{U}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{x}^{\mathrm{n}} \mathrm{arctan}\left(\mathrm{x}\right)\mathrm{dx}\:\mathrm{with}\:\mathrm{n}\:\mathrm{integr}\:\mathrm{nstural} \\ $$

Question Number 124906 Answers: 0 Comments: 1

∫sinx^3 dx=?

$$\int\boldsymbol{{sinx}}^{\mathrm{3}} \boldsymbol{{dx}}=? \\ $$

Question Number 124903 Answers: 1 Comments: 0

Question Number 124888 Answers: 3 Comments: 0

::::: prove that :::: φ=∫_0 ^( ∞) ((arctan(x^2 ))/x^2 )dx=(π/( (√2)))

$$:::::\:\:{prove}\:{that}\: \\ $$$$\:\:::::\:\:\:\:\:\:\phi=\int_{\mathrm{0}} ^{\:\infty} \frac{{arctan}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }{dx}=\frac{\pi}{\:\sqrt{\mathrm{2}}} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\: \\ $$

Question Number 124887 Answers: 1 Comments: 0

...nice calculus.. evaluate : 2∫_1 ^( ∞) ((({x}−(1/2))/x))dx−∫_0 ^( 1) ln(Γ(x))dx=??? {x}: fractional part...

$$\:\:\:\:\:...{nice}\:\:{calculus}.. \\ $$$$\:\:\:{evaluate}\:: \\ $$$$\:\:\mathrm{2}\int_{\mathrm{1}} ^{\:\infty} \left(\frac{\left\{{x}\right\}−\frac{\mathrm{1}}{\mathrm{2}}}{{x}}\right){dx}−\int_{\mathrm{0}} ^{\:\mathrm{1}} {ln}\left(\Gamma\left({x}\right)\right){dx}=??? \\ $$$$\left\{{x}\right\}:\:{fractional}\:{part}... \\ $$

Question Number 124827 Answers: 2 Comments: 0

.... nice calculus ... prove that:: ∫_0 ^( (π/2)) ((log(1+tan(x)))/(tan(x)))dx=((5π^2 )/(48)) ✓

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:....\:{nice}\:\:\:{calculus}\:... \\ $$$$\:\:\:\:\:\:\:{prove}\:{that}:: \\ $$$$\:\:\:\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{{log}\left(\mathrm{1}+{tan}\left({x}\right)\right)}{{tan}\left({x}\right)}{dx}=\frac{\mathrm{5}\pi^{\mathrm{2}} }{\mathrm{48}}\:\checkmark \\ $$$$ \\ $$

Question Number 124825 Answers: 0 Comments: 2

Question Number 124785 Answers: 0 Comments: 0

∫_( 0) ^( a) ∫_( 0) ^( (√(a^2 −x^2 ))) (1/((1+e^y )(√(a^2 −x^2 −y^2 ))))dxdy

$$\int_{\:\mathrm{0}} ^{\:\mathrm{a}} \int_{\:\mathrm{0}} ^{\:\sqrt{\mathrm{a}^{\mathrm{2}} −\mathrm{x}^{\mathrm{2}} }} \frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{e}^{\mathrm{y}} \right)\sqrt{\mathrm{a}^{\mathrm{2}} −\mathrm{x}^{\mathrm{2}} −\mathrm{y}^{\mathrm{2}} }}\mathrm{dxdy} \\ $$$$ \\ $$

Question Number 124794 Answers: 1 Comments: 0

If f(x)= { ((2x ; 0<x<1)),((3 ; x=1 )),((6x−1 ; 1<x<2)) :} find ∫_0 ^2 f(x) dx ?

$${If}\:{f}\left({x}\right)=\begin{cases}{\mathrm{2}{x}\:;\:\mathrm{0}<{x}<\mathrm{1}}\\{\mathrm{3}\:;\:{x}=\mathrm{1}\:}\\{\mathrm{6}{x}−\mathrm{1}\:;\:\mathrm{1}<{x}<\mathrm{2}}\end{cases} \\ $$$${find}\:\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\:{f}\left({x}\right)\:{dx}\:? \\ $$

Question Number 124853 Answers: 2 Comments: 0

∫_1 ^( x^3 +5x) f(t) dt = 2x then f(18) =?

$$\:\:\int_{\mathrm{1}} ^{\:{x}^{\mathrm{3}} +\mathrm{5}{x}} {f}\left({t}\right)\:{dt}\:=\:\mathrm{2}{x}\:\: \\ $$$$\:{then}\:{f}\left(\mathrm{18}\right)\:=? \\ $$

Question Number 124742 Answers: 0 Comments: 2

∫((3^t +11)/(6^t +11))dt collected problem

$$\int\frac{\mathrm{3}^{{t}} +\mathrm{11}}{\mathrm{6}^{{t}} +\mathrm{11}}{dt}\:\:\:\boldsymbol{{collected}}\:\boldsymbol{{problem}} \\ $$

Question Number 124738 Answers: 0 Comments: 3

...nice calculus... simple limit:: lim_(n→∞) {((1^(a+1) +2^(a+1) +...+n^(a+1) )/(n(1^a +2^a +....n^a )))}=? where a ≠−2 , −1

$$\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:\:{calculus}... \\ $$$$\:{simple}\:{limit}:: \\ $$$$\:\:\:{lim}_{{n}\rightarrow\infty} \:\left\{\frac{\mathrm{1}^{{a}+\mathrm{1}} +\mathrm{2}^{{a}+\mathrm{1}} +...+{n}^{{a}+\mathrm{1}} }{{n}\left(\mathrm{1}^{{a}} +\mathrm{2}^{{a}} +....{n}^{{a}} \right)}\right\}=? \\ $$$$\:{where}\:{a}\:\neq−\mathrm{2}\:,\:−\mathrm{1} \\ $$

Question Number 124675 Answers: 0 Comments: 1

∫_1 ^3 (x−1)^3 (3−x)^2 dx

$$\int_{\mathrm{1}} ^{\mathrm{3}} \left({x}−\mathrm{1}\right)^{\mathrm{3}} \left(\mathrm{3}−{x}\right)^{\mathrm{2}} {dx} \\ $$

Question Number 124672 Answers: 1 Comments: 1

∫_(−3) ^(−2) (y+3)^6 (y+2)^4 dy

$$\int_{−\mathrm{3}} ^{−\mathrm{2}} \left({y}+\mathrm{3}\right)^{\mathrm{6}} \left({y}+\mathrm{2}\right)^{\mathrm{4}} {dy} \\ $$

Question Number 124654 Answers: 3 Comments: 1

∫_(1/(√2)) ^(1/2) (e^(cos^(−1) (x)) /( (√(1−x^2 )))) dx ?

$$\:\underset{\mathrm{1}/\sqrt{\mathrm{2}}} {\overset{\mathrm{1}/\mathrm{2}} {\int}}\:\frac{{e}^{\mathrm{cos}^{−\mathrm{1}} \left({x}\right)} }{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}\:{dx}\:?\: \\ $$

Question Number 124644 Answers: 2 Comments: 0

∫_(2/(√3)) ^2 ((cos (sec^(−1) x))/(x(√(x^2 −1)))) dx ∫_( (√2)) ^2 ((sec^2 (sec^(−1) x))/(x(√(x^2 −1)))) dx

$$\underset{\mathrm{2}/\sqrt{\mathrm{3}}} {\overset{\mathrm{2}} {\int}}\:\frac{\mathrm{cos}\:\left(\mathrm{sec}^{−\mathrm{1}} {x}\right)}{{x}\sqrt{{x}^{\mathrm{2}} −\mathrm{1}}}\:{dx}\: \\ $$$$\underset{\:\sqrt{\mathrm{2}}} {\overset{\mathrm{2}} {\int}}\:\frac{\mathrm{sec}\:^{\mathrm{2}} \left(\mathrm{sec}^{−\mathrm{1}} {x}\right)}{{x}\sqrt{{x}^{\mathrm{2}} −\mathrm{1}}}\:{dx}\: \\ $$

Question Number 124632 Answers: 3 Comments: 0

Calculate ∫_0 ^2 (√((2+x)/(2−x))) dx

$${Calculate}\:\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\:\sqrt{\frac{\mathrm{2}+{x}}{\mathrm{2}−{x}}}\:{dx}\: \\ $$

Question Number 124624 Answers: 1 Comments: 0

∫ (dx/( (x)^(1/3) +4x))

$$\:\int\:\frac{{dx}}{\:\sqrt[{\mathrm{3}}]{{x}}\:+\mathrm{4}{x}} \\ $$

Question Number 124608 Answers: 2 Comments: 0

∫_0 ^∞ sinx^p dx ∫_0 ^∞ ((sinx^p )/x^q )dx collected question

$$\int_{\mathrm{0}} ^{\infty} {sinx}^{{p}} \:{dx} \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{{sinx}^{{p}} }{{x}^{{q}} }{dx} \\ $$$${collected}\:{question} \\ $$

Question Number 124607 Answers: 2 Comments: 0

∫(dx/((x^3 +1)^2 )) = ?

$$\int\frac{\boldsymbol{{dx}}}{\left(\boldsymbol{{x}}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{2}} }\:=\:? \\ $$

Question Number 124594 Answers: 1 Comments: 1

Show that ∫_0 ^(ln 2) (1/(cosh(x + ln 4))) dx = 2 tan^(−1) ((4/(33)))

$$\mathrm{Show}\:\mathrm{that}\: \\ $$$$\:\:\:\int_{\mathrm{0}} ^{\mathrm{ln}\:\mathrm{2}} \frac{\mathrm{1}}{\mathrm{cosh}\left({x}\:+\:\mathrm{ln}\:\mathrm{4}\right)}\:{dx}\:=\:\mathrm{2}\:\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{4}}{\mathrm{33}}\right) \\ $$

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