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:
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.
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
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.
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,
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?