hw-06-2-volumes - called"cubic...

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Homework for Chapter 6.2: Volumes For the problems that involve a little bit of numeric computation, feel free to use Excel or some other similar program. As before, you need only turn in the problems that have no answer in the back of the book. Drill problems: 2 y=1-x^2, y=0; about the x-axis 5 x=2sqrt(y), x=0, y=9, about the y-axis 8 y=0.25 x^2, y=5-x^2; about the x-axis 14 y=1/x, y=0, x=1, x=3, about y=-1 16 y=x, y=sqrt(x); about x=2 32 set up y=(x-2)^4, 8x-y=16; about x=10 34 set up: y=0, y=sin(x), x between 0 and pi; about y=-2 36 y=cos x, y=2-cos x, x between 0 and 2pi; about y=4 42 what solid has volume pi*int_2^5 y dy ? Applied problems: 45 CAT scan 46 lumber log 47 volume from graph 65 Cavalieri's principle: just read the question. Question A: Consider the ellipse (x/a)^2 + (y/b)^2 = 1 and rotate it around the x-axis. What is the resulting volume? You will get a formula in terms of pi, a, and b. Question B: (optional) It is quite common in industry to represent a shape by using cubic polynomials,
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Unformatted text preview: called "cubic splines". i) Find a generic formula for the volume generated by spinning the area under a x^3 + b x^2 + c x + d from x=m to x=n, around the x-axis. ii) Adapt your formula (without re-doing any integration) to the polynomial a (x-m)^3 + b (x-m)^2 + c (x-m) + d from x=m to x=n, around the x-axis. Challenge (optional/extra credit) problems: 63 volume of a torus 65 Cavalieri's principle: do it, rather than just reading it. 66 intersecting circular cylinders 70 circular hole in a sphere 72 switching from around x-axis to around a lower horizontal line---------------------------When we get toward the end of Chapter 6, we will work on these Problems Plus problems: (pg 448) #4 tilted cylinder w/water #5a segment of a sphere, c) depth of floating sphere #6 Archimedes principle #7 evaporation, depth of water decreases at a constant rate So, go read them now. Some of them you can even solve now! But they aren't due yet....
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This note was uploaded on 09/08/2011 for the course MATH 121 at Eastern Michigan University.

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