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Thread: HELP!! Volume of a solid w/ uniform cross sections of a parabola

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    HELP!! Volume of a solid w/ uniform cross sections of a parabola

    I didn't know how to explain the problem in words so I made a scan of it.




    Edit: Any help at all will be very appreciated.
    Last edited by sj9110; Dec 9th 2008 at 06:37 AM.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by sj9110 View Post
    I need to find the volume of a parabola a-x^2 where a is 0 to 10 and its also bounded by y=x or something. I don't know. Clearly I am very confused. I have a drawing of what the solid should look like.

    I didn't know how to explain the problem in words so I made a scan of it.




    Any help at all will be very appreciated.
    recall that the volume between $\displaystyle x = a$ and $\displaystyle x = b$ ($\displaystyle a \le b$) is given by $\displaystyle V = \int_a^b A(x)~dx$, where $\displaystyle A(x)$ is the formula for the cross-sectional area.

    in this case, our cross-sections are parabolas of the form $\displaystyle y = a - x^2$. but what is the area of this parabola that fits your constraints? recall that the area "under" the curve is given by the integral. so graph the curve $\displaystyle y = a - x^2$ for arbitrary $\displaystyle a$. find the intercepts and all that. now find the area under this curve. we see the area will be given by: $\displaystyle A = \int_{\sqrt{a}}^{\sqrt{a}} (a - x^2)~dx = 2 \int_0^{\sqrt{a}}(a - x^2)~dx$.

    Now, when you get the integral, replace the a with x, since in this case, a = x (the parabola is bounded by y = x, so the height is given by x, so that a = x since a is the height of our parabola above the x-axis). then go to the volume formula i gave you above to find the answer. x ranges from 0 up to 10
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  3. #3
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    Solid with parabolic cross-section

    Hi -

    I attach a drawing of the problem as I understand it, with three axes: $\displaystyle Ox$, $\displaystyle Oy$ and $\displaystyle Oz$.

    The values of $\displaystyle x$ go from $\displaystyle 0$ to $\displaystyle 10$. At $\displaystyle x = a$, the parabola (which lies in a plane parallel to the $\displaystyle y-z$ plane) has equation $\displaystyle z=a-y^2$. This parabola will have values of $\displaystyle y$ from $\displaystyle -\sqrt a$ to $\displaystyle \sqrt a$, and values of $\displaystyle z$ from $\displaystyle 0$ to $\displaystyle a$.

    What you need to do, then, to find the volume of the solid enclosed by all these parabolas, the $\displaystyle x-y $ plane and the plane $\displaystyle x = 10$, is:

    • Find the area $\displaystyle A$ of the typical parabola shown. (Do this in the usual way with an integral, whose limits are $\displaystyle y=-\sqrt a$ to $\displaystyle y=\sqrt a$.)
    • Replace $\displaystyle a$ by $\displaystyle x$ in your formula for $\displaystyle A$.
    • Now imagine $\displaystyle x$ increasing by an amount $\displaystyle \delta x$. As it does so, the volume 'swept out' is approximately $\displaystyle A\delta x$. So the total volume will be $\displaystyle \int_0^{10} A dx$. So, replace $\displaystyle A$ by your formula in terms of $\displaystyle x$, and then work out the integral.

    Have I given you enough to go on? Let me know if you want me to check your working.

    Grandad
    Attached Thumbnails Attached Thumbnails HELP!! Volume of a solid w/ uniform cross sections of a parabola-paraboliccrosssection.jpg  
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Grandad View Post
    Hi -

    I attach a drawing of the problem as I understand it, with three axes: $\displaystyle Ox$, $\displaystyle Oy$ and $\displaystyle Oz$.

    The values of $\displaystyle x$ go from $\displaystyle 0$ to $\displaystyle 10$. At $\displaystyle x = a$, the parabola (which lies in a plane parallel to the $\displaystyle y-z$ plane) has equation $\displaystyle z=a-y^2$. This parabola will have values of $\displaystyle y$ from $\displaystyle -\sqrt a$ to $\displaystyle \sqrt a$, and values of $\displaystyle z$ from $\displaystyle 0$ to $\displaystyle a$.

    What you need to do, then, to find the volume of the solid enclosed by all these parabolas, the $\displaystyle x-y $ plane and the plane $\displaystyle x = 10$, is:

    • Find the area $\displaystyle A$ of the typical parabola shown. (Do this in the usual way with an integral, whose limits are $\displaystyle y=-\sqrt a$ to $\displaystyle y=\sqrt a$.)
    • Replace $\displaystyle a$ by $\displaystyle x$ in your formula for $\displaystyle A$.
    • Now imagine $\displaystyle x$ increasing by an amount $\displaystyle \delta x$. As it does so, the volume 'swept out' is approximately $\displaystyle A\delta x$. So the total volume will be $\displaystyle \int_0^{10} A dx$. So, replace $\displaystyle A$ by your formula in terms of $\displaystyle x$, and then work out the integral.

    Have I given you enough to go on? Let me know if you want me to check your working.

    Grandad
    this is what i mean, i answered this problem in the other thread, giving the same solution (though i must admit, Grandad was a bit more elegant, using 3-dimensions and giving a nice diagram to boot--but it was still the same solution).
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