Question

In Exercises 13-16, evaluate \(\int_{S} \int f(x, y) d S\). $$ \begin{aligned} &f(x, y)=y+5\\\ &S: \mathbf{r}(u, v)=u \mathbf{i}+v \mathbf{j}+\frac{v}{2} \mathbf{k}, \quad 0 \leq u \leq 1, \quad 0 \leq v \leq 2 \end{aligned} $$

Short answer

The value of the given double integral is 6.

Step-by-step solution

Conversion to Parametric Form

Start by writing the function \(f(x, y)\) in the form of \(f(u, v)\) by substituting \(x=u\) and \(y=v\). Hence, \(f(u, v) = v + 5\).

Calculation of the Jacobian

The Jacobian determinant of the transformation from rectangular to parametric coordinates is computed from \(\frac{\partial(x,y,z)}{\partial(u,v)}\). Here, \(\frac{\partial(x,y,z)}{\partial(u,v)} = \begin{vmatrix}1 & 0\ 0 & 1\ 0 & \frac{1}{2}\end{vmatrix}\) = 0.5.

Computation of the Double Integral

The transformation of the double integral over \(S\) into a double integral over \(R\) involves replacing \(dS\) with the absolute value of the Jacobian determinant \(|J|\) times \(dudv\). The double integral over \(R\) to be evaluated is thus \(\int_{0}^{1} \int_{0}^{2} (v +5 ) * 0.5 dudv\).

Calculation of the Inner Integral

Evaluate the inner integral with respect to \(u\): \(\int_{0}^{1} (0.5v + 2.5) du\) = \([0.5vu + 2.5u]_0^1\) = \(0.5v + 2.5\).

Calculation of the Outer Integral

Evaluate the outer integral with respect to \(v\): \(\int_{0}^{2} (0.5v +2.5) dv\) = \([0.25v^2 + 2.5v]_0^2\) = \(0.25*4 + 2.5*2\) = \(1+5\) = \(6\).