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Math Help - Help Please need Clarfication Integral Volumes.!

  1. #1
    Member zangestu888's Avatar
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    Help Please need Clarfication Integral Volumes.!

    The question is;
    Find the volume of the solid whose base is the region bounded by
    y = e^x, y = 1, x = 2 and x = 3 ,

    where cross-sections perpendicular to the
    x-axis are semicircles.

    The part of this equation I do not understand is the set up, am not sure how all y=1 comes into play? and how i should revolve this graph to get a solid of revoultion? basically dont know were to start? If anyone please help me with a starting point thank you .
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  2. #2
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    skeeter's Avatar
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    Quote Originally Posted by zangestu888 View Post
    The question is;
    Find the volume of the solid whose base is the region bounded by
    y = e^x, y = 1, x = 2 and x = 3 ,

    where cross-sections perpendicular to the
    x-axis are semicircles.

    The part of this equation I do not understand is the set up, am not sure how all y=1 comes into play? and how i should revolve this graph to get a solid of revoultion? basically dont know were to start? If anyone please help me with a starting point thank you .
    this is not a solid of revolution ... it is a solid formed by similar cross-sections.

    V = \int_a^b A(x) \, dx

    in this case, each similar cross section is a semicircle. the diameter of a semicircular cross-section is d = e^x - 1 , so, the radius of a semicircular cross-section is r = \frac{e^x-1}{2}

    therefore, the representative cross sectional area as a function of x is

    A(x) = \frac{1}{2} \pi \left(\frac{e^x-1}{2}\right)^2

    limits of integration will be from x = 2 to x = 3.

    you have all the information you need ... set up the volume integral and evaluate.
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  3. #3
    Member zangestu888's Avatar
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    Thanks :)

    So d = e^x - 1, the -1 comes from the fact that the region has the line y=1 correct, that was the part that was really the most confusing I just cant picture all of this sometimes, is thier a trick lol thanks! much appreciated
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  4. #4
    Senior Member mollymcf2009's Avatar
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    Quote Originally Posted by zangestu888 View Post
    The question is;
    Find the volume of the solid whose base is the region bounded by
    y = e^x, y = 1, x = 2 and x = 3 ,

    where cross-sections perpendicular to the
    x-axis are semicircles.

    The part of this equation I do not understand is the set up, am not sure how all y=1 comes into play? and how i should revolve this graph to get a solid of revoultion? basically dont know were to start? If anyone please help me with a starting point thank you .

    First you need to draw the graph of what this looks like. The region bounded by these curves is above y=1, below y=e^x and between x=1 & x=2. This is the area that you will be finding the volume of.

    I assume you have learned how to use estimating rectangles? Draw your estimating rectangle from y =e^x to y=1. This is the section that if you moved it back and forth would create semi-circles perpendicular to the x-axis, it would NOT be a full revolution! .

    Does that help you get started?
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