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Math Help - Volume of revolution problem

  1. #1
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    Exclamation Volume of revolution problem

    Hi everyone, I have the following problem: What is the solid formed by rotating the region bounded by y=ln(x+1),x=0,y=2 around the y axis? The answer would be its pi/2(e^4-4e^2+7). How would one solve this problem? Would one find the intersection of y=2 and y=ln(x+1) which i got to be ((e^2)-1,ln(e^2)) and then take the integral of ln(x+1) from o to (e^2)-1 away from the area under y=2 from o to (e^2)-1 to get formula the cross section and then to find the volume integrate the formula for the cross section squared (radius) from y=0 to 2 and then times by pi?

    Is this correct?

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  2. #2
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    Re: Volume of revolution problem

    First of all, surely you can simplify  \displaystyle \begin{align*} \ln{\left( e^2 \right)} \end{align*} ...
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  3. #3
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    Re: Volume of revolution problem

    So my point of intersection is ((e^2)-1,2) but is the process correct?

    Thanks for the reply
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  4. #4
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    Re: Volume of revolution problem

    First, graph the region to be revolved:

    Volume of revolution problem-logrev.jpg

    Next, compute the volume of an arbitrary disk:

    dV=\pi r^2\,dy

    r=x=e^y-1

    Hence:

    dV=\pi (e^y-1)^2\,dy

    Finally sum up the disks by integrating:

    V=\pi\int_0^2(e^y-1)^2\,dy
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  5. #5
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    Re: Volume of revolution problem

    How did you get the radius = e^y-1?
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  6. #6
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    Re: Volume of revolution problem

    Because the function is  \displaystyle \begin{align*}  y = \ln{(x + 1)} \end{align*} and the radius of your solid is given by how far horizonally you extend, so  \displaystyle \begin{align*} x \end{align*} . Surely you can see that if  \displaystyle \begin{align*} y = \ln{(x + 1)} \end{align*} then  \displaystyle \begin{align*} x = e^y - 1 \end{align*} ...
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  7. #7
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    Re: Volume of revolution problem

    Im sorry i cannot see that could you please explain? You extend to the x coordinate of intersection dont you which is (e^2)-1.
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  8. #8
    MHF Contributor MarkFL's Avatar
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    Re: Volume of revolution problem

    No, for an arbitrary disk, the radius is found by extending from the y-axis to the x-coordinate on the logarithmic curve.

    Look at the graph I provided and picture an arbitrary horizontal line in the shaded area serving as the radius of some disk.
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