1. Volume and Work A container is created by revolving the curve y=x2″ title=”y equals x squared” src=”https://lh4.googleusercontent.com/WJPq_KVsGG8UUgPyELPVR7K5yLLkZfAm8KUf4OYV16GLWt1OZoUFqwyJI2m75-v

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1. Tome and Work

A container is created by revolving the deflexion y=x2" title="y equals x squared" src="https://lh4.googleusercontent.com/WJPq_KVsGG8UUgPyELPVR7K5yLLkZfAm8KUf4OYV16GLWt1OZoUFqwyJI2m75-vuzXeHFl71y2XxS78FB-vUZ25ZRIjc7IY72TZlCGcJ3koGht-qzIBIK8yOi_isQs9-2oB86HxT" width="54" elevation="23" style="margin-left: 0px; margin-top: 0px;">  from y=0 to y= 9 environing the y-axis. 

a. Write an sound that computes V(h), the tome of mellifluous contained if the container is occupied to a elevation h.

b. How ample insinuate does this container dwell when it is generous?

c. To what elevation does the insinuate flatten aim when the tome is half-full?

d. If the container is generous of insinuate, how ample operation does it interest to cross-examine all of the insinuate out of the container?  Use the symbols ρ" title="rho" src="https://lh4.googleusercontent.com/uSmw8P3W7ReX7fXmU4iju7SdXc1vToia52ydLqgXKn8lWRIIDuOQUe90m5zeVZMp9_Xti3Jda1-g0vz6NcYH2eMMw80F_ijjIWWQuwTIMc8VfhWbovlTJzcYhnHHuFXOxLALoPZw" width="15" elevation="19" style="margin-left: 0px; margin-top: 0px;"> and g in your computations to indicate dullness of insinuate and aid of lugubriousness.  If you’d affect a calculate at the end, you can estimateρ=1000kg/m3" title="rho equals 1000 k g slash m cubed" src="https://lh4.googleusercontent.com/dYMhvk_WoQm38ZfbxpoH_zU4yXp0WFBF_ImZvh-Uh6TrbUcvdPEsXYMsyXpfLrdqYEEn4nkh68EU7iCYFIy8e1HpyZjwfA9htUsxpYM6BlErHHvKTecLbjuF2qB0Z_Ra3DOGeCbq" width="129" elevation="24" style="margin-left: 0px; margin-top: 0px;"> and g=10m/s2" title="g equals 10 m slash s squared" src="https://lh6.googleusercontent.com/MPElEv8zC5RIFTj3plVNsWGqbBG86fyQV0vIBAYg1Hn-jVG-JNkxd_Qob8AO2T6K7qnTxXCEST-Rc3CaG04bp4HPUlAtlnzqIb-TfR-c9YuBhabDdqUaD-GGeSTEDMmcDQFBniqZ" width="96" elevation="24" style="margin-left: 0px; margin-top: 0px;">

e. Your cross-examine breaks down succeeding cross-examineing out half of the tome of insinuate in the tank.  What uniformity of the operation required to cross-examine all the insinuate out was effected?  (Hint: use your solution from aloft, and believe environing cross-examineing the insinuate out from the top of the tank down to the retaining insinuate flatten.)

2. Integration by Parts

Let p(x) be an imageless office defined on the intermission [a, b] after a while the subjoined properties:

p(a) = p(b) = 0

p’’(x) exists for each x in [a,b].

a. Use these properties to semblance that ∫abp(x)p″(x)dx=−∫ab(p′(x))2dx" title="the sound from a. to b of p of x p envelop superexcellent of x equals disclaiming the sound from a. to b of notorious paren p superexcellent of x delay paren squared" src="https://lh6.googleusercontent.com/6i1MWXbQD5suOEkJC1N8AnKOMRH9Mn2IFuBNTPAi_SpcR4nLUIXirQKd-1v1uC4E7T9BeILcD1HQSqJPElG8f-2gObJCj4QxKCC57GqCaZSy2fg7PEFxVavtND-wYu56ApNjBQbf" width="336" elevation="56" style="margin-left: 0px; margin-top: 0px;">.  (Hint, complete the left party by faculty and elucidate).

As another party hush, it allure be profitable to use the integration by faculty formula if the sounds keep limits:

∫abf(x)g′(x)dx+∫abf′(x)g(x)dx=f(x)g(x)|ab" title="the sound from a. to b of f of x g superexcellent of x plus the sound from a. to b of f superexcellent of x g of x equals f of x g of x divides sub a. to the bth power" src="https://lh3.googleusercontent.com/f-nKprXTAFJZBUufwVMv-_VOGaL-sQ6QclY4lSLgGRRl35e1QjvbMnN7U3tklpJMKdWK4LX9vNlgirQqOiKSSB0r-XjXLB5y8X-yIuY04vyJLbCRrHKO3t4OXhQeAjSyWNLfznpr" width="455" elevation="56" style="margin-left: 0px; margin-top: 0px;">.    Hush that the exact party of the equation is evaluated from a to b.

b. In specification to the properties aloft, judge p’’ is uniformityal to p.  In symbols. p’’(x) = Kp(x), for some true K.  Semblance that K must be a disclaiming calculate.

c. We allure allot these results to operation on a dishonorable sound next week.  For now, we righteous believe environing offices that recompense each of the subjoined criteria:

Make up two offices, along after a while an intermission [a, b] for each so that f(a) = f(b) = 0.

Make up two divergent offices so that f’’(x) = K f(x) for some true K.

For each of the disgusting offices you wrote down, proof each term.


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