# Feb 5.pdf

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Work and Fluid Forces 6.5!
If a variable F(x) is causing an object to move from x=a to x=b, then the work can
be approximated by a Riemann sum:!
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The exact work, obtained as the limit of Riemann sums, is given by!
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Example pg 396 #5!
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The conical tank is ﬁlled to within 2 ft of the top with olive oil weighing 57 lb/ft .
how much work does it take to pump the oil to the rim of the tank?!
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the precise amount of work is !
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Chapter 8 : techniques of Integration!
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Integration by parts 8.1!
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Recall the product rule (for differentiation):!
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It follows that!
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This is the integration by parts formula!
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Remark:!
for deﬁnite integrals, we have the following formula,!
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Example!
the graph of !
revolves around the x axis. ﬁnd the volume of the resulting solid.!
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Solution!
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