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Thread: Integrating a volume

  1. #1
    Junior Member
    Sep 2009

    Integrating a volume

    We just started today and a problem I have is...

    Write a Riemann sum and then a definite integral representing the volume of the region, using the slice shown in the figure below. Evaluate the integral exactly. Use your work to answer the questions below.

    What is the approximate volume of the slice with respect to y?

    So I got l/14=(7-y)/7 and then I get 20y deltay is the volume of the slice but it is a circle, where would I put pi into this?

    Thanks for the help,

    Attached Thumbnails Attached Thumbnails Integrating a volume-53286.jpg  
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  2. #2
    DBA is offline
    May 2009
    The area of the cut is a rectangle A=2x*10
    Radius is 7 so x=7
    <br />
V = \int_{a}^{b}A(y)dy<br />

    In your case you have

    <br />
V = \int_{a}^{b} 2x * 10 dy = 20 \int_{a}^{b} x dy <br />

    So, you need to write the x in terms of y, since you have dy

    The point (x,y) is on the circel
    x^{2} + y^{2} = r^{2}
    x^{2} + y^{2} = 7^{2}

     x = \sqrt {49-y^{2}}

    <br />
V = 20 \int_{0}^{7} \sqrt {49-y^{2}} dy <br />
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