i need help with two questions for my calculus..

label Calculus
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Apr 27th, 2015

first problem

This is relatively easy to do by the method of disks, where a typical disk has radius x(k-x) and width dx. Thus the volume of the solid of revolution is given by
V = pi int_{x=0}^{x=k} x^2(k-x)^2 dx
= pi int_{x=0}^{x=k} x^2(k^2 - 2kx + x^2) dx
= pi [k^2(x^3/3) - 2k(x^4/4) + x^5/5]_{x=0}^{x=k}
= pi [k^5/3 - k^5/2 + k^5/5]
= pi(k^5/30)[10 - 15 + 6]
= pi(k^5/30).


second problem

you could probably do this by the method of disks, but the radii of the outer and inner disks in this case are rather awkward functions involving the (two) solutions of x(k-x) = y (in terms of x). So let's use the method of shells instead, where the typical shell has radius x and height x(k-x). Then the volume of the solid of revolution is given by

V = 2pi int_{x=0}^{x=k} x^2(k-x) dx
= 2pi [kx^3/3 - x^4/4]_{x=0}^{x=k}
= 2pi [k^4/3 - k^4/4]
= pi(k^4)/6.

Apr 27th, 2015

what about the third one


Apr 28th, 2015

can you write it by hand and upload as pdf

Apr 28th, 2015

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Apr 27th, 2015
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Apr 27th, 2015
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