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Thread: Integration to Find Volume of Wine Glass

  1. #1
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    Exclamation Integration to Find Volume of Wine Glass

    Integration to Find Volume of Wine Glass-evernote-snapshot-20150826-130914.jpg

    Is anyone able to help me set out the algebra/graph to solve this?

    Thanks,
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  2. #2
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    Re: Integration to Find Volume of Wine Glass

    The volume of a region revolved about the x-axis is given by

    $V=\pi \displaystyle{\int_a^b}[f(x)]^2~dx$

    In this case

    $f(x) = \sqrt{x}$

    $a=0$

    $b=?$

    We want to find $b$ in order for the glass to hold 100 ml. Note that $1ml = {1cm}^3$ so if we measure $b$ in $cm$ we want a volume of $100$

    This is a pretty trivial integral. See if you can finish from here.

    Spoiler:
    $V=\dfrac {\pi b^2}{2}=100$

    $b=\sqrt{\dfrac{200}{\pi}}=10\sqrt{\dfrac{2}{\pi}} $

    we ignore the solution where $b<0$

    so $b \approx 8cm$
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  3. #3
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    Exclamation Re: Integration to Find Volume of Wine Glass

    I worked that part out thanks to your help! Are you able to help me with another part too? I have posted the question and graph below.
    Integration to Find Volume of Wine Glass-screen-shot-2015-09-01-3.07.48-pm.png Integration to Find Volume of Wine Glass-screen-shot-2015-09-01-3.08.18-pm.jpg
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  4. #4
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    Re: Integration to Find Volume of Wine Glass

    first thing is that you should rotate the image 90 clockwise so that this mimics the previous problem.

    Then using some photo software or the Mark I eyeball pick about a dozen or so points on that curve.

    You should be able to model it pretty accurately as a 3rd order polynomial so do so.

    Apply the technique of the previous problem to find the volume.
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  5. #5
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    Re: Integration to Find Volume of Wine Glass

    I've rotated the image but i'm not sure how to model it after picking points on the curve?
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  6. #6
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    Re: Integration to Find Volume of Wine Glass

    Quote Originally Posted by siddivi View Post
    I've rotated the image but i'm not sure how to model it after picking points on the curve?
    suppose you've got a set of points $(x_k,y_k),~~k=1,n$

    you want to model this as a $nth$ degree polynomial $\displaystyle{\sum_{k=0}^n}c_k x^k$

    let

    $X_{m,n} = x_m^n$

    $Y_{m,1}=y_m$

    $C_{m,1}=c_m$

    the fit to the polynomial can be written as

    $Y=XC$

    this in general will be overdetermined so you find a least squares fit by solving

    $X^TY=X^TXC$

    or

    $C=(X^TX)^{-1}X^TY$


    Least Squares Fitting--Polynomial -- from Wolfram MathWorld
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  7. #7
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    Re: Integration to Find Volume of Wine Glass

    Are you able to give me an example? I got lost eight the algebra above
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  8. #8
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    Re: Integration to Find Volume of Wine Glass

    suppose you had 4 data points

    $(x_1,y_1), \dots, (x_4,y_4)$

    form the matrix

    $X=\begin{pmatrix}1 &x_1 &x_1^2 &x_1^3 \\ 1 &x_2 &x_2^2 &x_2^3 \\ 1 &x_3 &x_3^2 &x_3^3 \\ 1 &x_4 &x_4^2 &x_4^3 \end{pmatrix}$

    form the vector

    $Y=\begin{pmatrix}y_1 \\ y_2 \\ y_3 \\ y_4 \end{pmatrix}$

    compute

    $C=\left(X^T X\right)^{-1}X^T Y$

    the elements of $C$ are the coefficients of your estimated 3rd order polynomial.
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  9. #9
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    Re: Integration to Find Volume of Wine Glass

    Sorry, my mathematical understanding isn't the best.

    If I have this set of points on the curve, how would i find the function?

    A (0,0)
    B (0.2, 0.9)
    C (0.8, 1.6)
    D (1.6, 2.3)
    E (2.7, 2.9)
    F (4.2,3.1)
    G (5.3, 3.2)
    H (6.3, 3.2)
    I (7.2, 3.1)
    J (7.8, 3.0)
    K (8.7, 2.8)
    L (9.5, 2.6)
    M (10.0, 2.4)

    Thanks,
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  10. #10
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    Re: Integration to Find Volume of Wine Glass

    I've given you enough help on this. You're just trying to get me to do your work for you now. I'm done.
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  11. #11
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    Re: Integration to Find Volume of Wine Glass

    Sorry, i'm just struggling to understand. In you post with the matrices, are you able to tell me what defines 'T' and what you the data in the 3rd and 4th columns of the X-Matrix?
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  12. #12
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    Re: Integration to Find Volume of Wine Glass

    And is 'T' the transition matrix?
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  13. #13
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    Re: Integration to Find Volume of Wine Glass

    T means the transpose of the matrix.

    The (m,n)th element of X, i.e. the number at the mth row and nth column, is the mth data point x value raised to the nth power.

    I really can't make it any clearer than in post #8.

    How did your professor expect to you do this polynomial modelling?
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