Integration along the side in isoparametric quadrilateral mapping

Question

I need to integrate a function (say,$F(x,y)=x^3+y^3$) along one side of the given quadrilateral. For example, the $ x $ and $ y$ coordinate of the quadrilateral are$ (0,0),(2,-1),(3,2),(1,3)$ and the concerned line is, the line joining $(2,-1)$ and $(3,2)$.

These are the steps which I followed,

1) INTEGRATING GEOMETRIC INTERPOLATION

1. Given quadrilateral is mapped into a standard square (of coordinates$(-1,-1),(1,-1),(1,1),(-1,1)$ in (s,t) ) by isoparametric mapping.
2. From Lagrangian interpolation $ X(s,t)= \sum N_ix_i$ and $Y(s,t)=\sum N_ix_i$ Where $N_1=(1-s)(1-t)/4,\ N_2=(1+s)(1-t)/4,\ N_3=(1+s)(1+t)/4 \ \ \ \ and \\ N_4=(1-s)(1+t)/4$ Hence, $$X(s,t)=(3(s + 1)(t + 1))/4 - ((s + 1)(t - 1))/2 - ((s - 1)(t + 1))/4 $$ $$Y(s,t)=((s + 1)(t - 1))/4 - (3(s - 1)*(t + 1))/4 + ((s + 1)(t + 1))/2$$
3. Along the given side $\ dl=\sqrt{dx^2+dy^2}$ and $ \ dl=\ \sqrt{(dX/dt)^2+(dY/dt)^2}\ \ dt$
4. Hence $$\int_{(2,-1)}^{(3,2)} x^3+y^3 \ dl =\int_{-1}^1 (X(1,t)^3+Y(1,t)^3) \sqrt{(dX/dt)^2+(dY/dt)^2}\ \ dt $$

Unfortunately, I get wrong integration result when I mapped my original quadrilateral into the standard square. Where did I make mistake in the integration?

2) LINE INTEGRAL CALCULATION $$\int_{(2,-1)}^{(3,2)}\ x^3+y^3 \ dl\ =\ \sqrt(10)\ \int_{0}^1\ {(m+2)}^3+(3m-1)^3 \ dm$$

3) INTEGRATING ELEMENT VARIABLE INTERPOLATION

1. $F_i(x_i,y_i)=\ x_i^3 +y_i^3$ is calculated at 4 points.
2. At any point in the standard square, function $F(x,y)$ is approximated as $$ \hat{F}(s,t)=\ \sum_{n=1}^4 N_iF_i(x_i,y_i)$$
   $N_i$ is Lagrangian shape function for standard square.
3. So the required integral is, $$\int_{(2,-1)}^{(3,2)}\ x^3+y^3 \ dl\ =\ \int_{-1}^1 \hat{F}(s,t) \sqrt{(dX/dt)^2+(dY/dt)^2}\ \ dt $$

RESULTS
The first and second methods gives the exact result $(55.3399)$ while third method gives $66.407$. The error in the third method is expected since the degree of interpolation function is less than that of real function.

PROBLEM
How do I get an accurate result from the integration of interpolation function of element variable (i.e by doing third method)? Is this related to subparametric mapping?