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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:!

!

!

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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Examples!

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1.!

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2.!

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3.!

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4.!

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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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