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Math Help - Volume of Revolution + Related Rates

  1. #1
    Member classicstrings's Avatar
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    Volume of Revolution + Related Rates

    We've just started these sort of questions, so I'm still trying to get the hang of them.

    a. The line with equation y = 2x, 0 <= x <= 2, is rotated about Y-axis. Find by integration the volume of the cone generated.

    b. Water is poured into the vessel above at a constant rate of 2m^3/min. At what rate is the water level rising when the depth is 1m. How long will it take to fill the vessel?

    c. Some time later, water flows out through the vertex of the cone and the depth decreases at a constant rate of c meters/min. At what rate does the surface area of the water decrease when the radius is 1m. How long does it take to empty the vessel?
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  2. #2
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    Quote Originally Posted by classicstrings View Post
    We've just started these sort of questions, so I'm still trying to get the hang of them.

    a. The line with equation y = 2x, 0 <= x <= 2, is rotated about Y-axis. Find by integration the volume of the cone generated.
    ...
    Hello,

    I'm in hurry therefore only

    a.

    Consider the cone constructed by small cylindrical solids. The radius of the cylinder is x:

    y = 2x \ \Longleftrightarrow \ r = x = \frac{1}{2}y

    and the height of the cylinder is dy.
    The volume of a cylinder is calculated by:

    V_{cyl} = \pi \cdot r^2

    Sum up all possible cylinders:

    V_{cone} = \int_{0}^4 \left(\pi \cdot \left(\frac{1}{2}y \right)^2 \right)dy =\pi \cdot  \left. \frac{1}{12}y^3 \right|_0^4 = \frac{16}{3}\pi
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