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Math Help - Volumes of solid of revolution, rotated about the y axis

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    Volumes of solid of revolution, rotated about the y axis

    I am not sure how to tackle this one.

    find the volume of y=x+1: x=0 and x=2, rotated about the y axis.
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    MHF Contributor FernandoRevilla's Avatar
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    Re: Volumes of solid of revolution, rotated about the y axis

    Quote Originally Posted by johnsy123 View Post
    I am not sure how to tackle this one.

    find the volume of y=x+1: x=0 and x=2, rotated about the y axis.
    Possibly that is not the exact formulation of the problem, the volume is not well defined.
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    Re: Volumes of solid of revolution, rotated about the y axis

    The previous question to this i had to work out the volume rotated about the x axis, this is where i got the points x=0 and x=2 from, but now they want me to find the volume of the solid about the yaxis. Should i sub the two x numbers into the function, to obtain y values? than work from there.
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    MHF Contributor Siron's Avatar
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    Re: Volumes of solid of revolution, rotated about the y axis

    I think that's a good solution, you have also to express now y in function of x, so: x=y-1 and then use the formula (in general):
    V=\pi \int_{a}^{b} [f(y)]^2dy
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    Re: Volumes of solid of revolution, rotated about the y axis

    Quote Originally Posted by johnsy123 View Post
    I am not sure how to tackle this one.

    find the volume of y=x+1: x=0 and x=2, rotated about the y axis.
    as stated by Fernando, the three given lines do not enclose a region in the plane.

    however, if you're speaking of the region in quadrant I bounded by the given lines as shown in the graph ...

    1. rotation about the x-axis

    V = \pi \int_0^2 (x+1)^2 \, dx

    2. rotation about the y-axis

    a. using the method of cylindrical shells w/r to x

    V = 2\pi \int_0^2 x(x+1) \, dx

    b. using the methods of disks and washers w/r to y (requires two integrals)

    V = \pi \int_0^1 2^2 \, dy + \pi \int_1^3 2^2 - (y-1)^2 \, dy

    c. finally, and probably the easiest, using geometry (cylinder volume - cone volume)

    V = \pi \cdot 2^2 \cdot 3 - \frac{\pi}{3} \cdot 2^2 \cdot 2
    Attached Thumbnails Attached Thumbnails Volumes of solid of revolution, rotated about the y axis-region.jpg  
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    Re: Volumes of solid of revolution, rotated about the y axis

    Why do you have to multiply by 2pi and not pi in previous cases?
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    Re: Volumes of solid of revolution, rotated about the y axis

    Quote Originally Posted by johnsy123 View Post
    Why do you have to multiply by 2pi and not pi in previous cases?
    Skeeter said he was using cylindrical shells. The surface area of a cylinder of radius r and height h is 2\pi rh, the circumference of the cylinder times the height of the cylinder.
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