Surface Integrals $\iint_S G(\mathbf{r}) d A$. Using $\iint_S G(\mathbf{r}) d A=\iint_R G(\mathbf{r}(u, v))|\mathbf{N}(u, v)| d u d v$ or $\iint_S G(\mathbf{r}) d A=\iint_{R^*} G(x, y, f(x, y)) \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2} d x d y$, evaluate thise integral for the given data. (Show the details.)

$$G=z, \quad S: x^2+y^2+z^2=9, \quad z \geqq 0$$

Find the equation for the inverse of the function $f ( x ) = 2 \sqrt { \frac { ( x - 3 ) } { 4 } } + 1$ Sketch the graph of both the original and the inverse.

For the following question,

Find the inverse of the function on the given domain.

$$f(x)=9-x^2,[0, \infty)$$

Using $\iint_R \nabla^2 w d x d y=\oint_C \frac{\partial w}{\partial n} d s$, evaluate $\oint_C \frac{\partial w}{\partial n} d s$ counterclockwise over the boundary curve $C$ of the region $R$. (Show the details of your work.)

$$w=3 x^2 y-y^3+y^2, \quad C: 25 x^2+y^2=25$$

Using $\iint_R \nabla^2 w d x d y=\oint_C \frac{\partial w}{\partial n} d s$, evaluate $\oint_C \frac{\partial w}{\partial n} d s$ counterclockwise over the boundary curve $C$ of the region $R$. (Show the details of your work.) $w=e^x \sin y, \quad R$ the triangle with vertices $(0,0),(4,2),(0,2)$

Surface Integral $\int_S \mathbf{F} \cdot \mathbf{n} d A$. Evaluate this integral by the divergence theorem for the following data. (Show the details.)

$$\mathbf{F}==\left[\begin{array}{ccc}\cos y, & \sin x, & \cos z\end{array}\right]$$

, $\quad S$ the surface of $x^2+y^2 \leqq 9, \quad 0 \leqq z \leqq 2$

Find a parametric representation of the following surface. (The answer gives one such representation and there are many others.) Find a normal vector. (Show the details.) Sphere $x^2+(y-1)^2+(z+2)^2=4$

Find the general solution of the differential equation $y''-3y'+2y=4x^2$

Find the general solution of the differential equation

$$2x^2y''-xy'-5y=6x^2$$

In the case of independence integrate from $(0,0,0)$ to $(a, b, c)$. (Show the details of your work.)

$$3(x+y)^2(d x+2 d y)+d z$$

Find the arc lengths.

$f(x) = \frac{1}{3}x^{3}+\frac{1}{4x}$ from $x=1$ to $x=2$

Find the exact length of the curve. y = 1/4x2 - 1/2 In x, 1≤ x ≤ 2

Evaluate $\int_C f(\mathbf{r}) d s$ with $f$ and $C$ as follows. (Show the details.) $f=\sqrt{2+x^2+3 y^2}, \quad C: \mathbf{r}=\left[t, t, t^2\right], 0 \leqq t \leqq 3$.

Find a solution of the differential equation x^2y'+xy=1 that satisfies the initial condition y(1)=2