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Math Help - Finding the volume of the sold in the first octant..

  1. #1
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    Finding the volume of the sold in the first octant..

    The question:
    Find the volume of the solid in the first octant bound by the graphs of z=1-y^2, y=2x, and x=3.

    My attempt:
    Finding the volume of the sold in the first octant..-daum_equation_1336349796440.png
    The reason why I changed the limits of integration was because I need x,y,z all to be greater than or equal to 0..however, I am not getting the correct answer of 15/8.
    Any suggestions?

    Thanks,
    Mike.
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  2. #2
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    Re: Finding the volume of the sold in the first octant..

    Your limits are off.

    It should be clear that

    \displaystyle \begin{align*} 0 \leq x &\leq 3 \\ \\ 0 \leq y &\leq 2x \\ \\ 0 \leq z &\leq 1 - y^2 \end{align*}


    So your integral should be

    \displaystyle \begin{align*} \int_{0}^{3}{\int_{0}^{2x}{\int_{0}^{1-y^2}{\,dz}\,dy}\,dx} \end{align*}
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  3. #3
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    Re: Finding the volume of the sold in the first octant..

    Quote Originally Posted by Prove It View Post
    Your limits are off.

    It should be clear that

    \displaystyle \begin{align*} 0 \leq x &\leq 3 \\ \\ 0 \leq y &\leq 2x \\ \\ 0 \leq z &\leq 1 - y^2 \end{align*}

    So your integral should be

    \displaystyle \begin{align*} \int_{0}^{3}{\int_{0}^{2x}{\int_{0}^{1-y^2}{\,dz}\,dy}\,dx} \end{align*}

    Thank you for your response.

    I originally had my limits set up that way, and my final answer was still off..as it was calculated to be -45.

    Maybe there is something wrong with my calculations..
    Finding the volume of the sold in the first octant..-daum_equation_1336362479173.png
    any thoughts?
    Thanks
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