In given problem, $R=\{(x, y): 0 \leq x \leq 6,0 \leq y \leq 4\}$ and P is the partition of R into six equal squares by the lines x=2, x=4, and y=2. Approximate $\iint_R f(x, y) d A$ by calculating the corresponding Riemann sum $\sum_{k=1}^6 f\left(\bar{x}_k, \bar{y}_k\right) \Delta A_k$, assuming that $\left(\bar{x}_k, \bar{y}_k\right)$ are the centers of the six squares.

$$f(x, y)=\frac{1}{6}(48-4 x-3 y)$$

In given problem, find the moments of inertia $I_x, I_y$, and $I_z$ for the lamina bounded by the given curves and with the indicated density $\delta$.

$$y=x^2, y=4 ; \delta(x, y)=y$$

Evaluate the iterated integrals in given problem.

$$\int_0^{\pi / 2} \int_0^{\sin y} e^x \cos y d x d y$$

In given problem, evaluate each of the iterated integrals.

$$\int_0^3 \int_0^1 2 x \sqrt{x^2+y} d x d y$$

In given problem, find the transformation from the uv-plane to the xy-plane and find the Jacobian. Assume that $x \geq 0$ and $y \geq 0$.

u=x+2 y, v=x-2 y

In given problem, sketch the solid S. Then write an iterated integral for

$$\iiint_S f(x, y, z) d V$$

$S=\{(x, y, z): 0 \leq x \leq 1,0 \leq y \leq 3$,

$$\left.0 \leq z \leq \frac{1}{6}(12-3 x-2 y)\right\}$$

In given problem, use cylindrical coordinates to find the indicated quantity.

Volume of the solid bounded above by the sphere $r^2+z^2=5$ and below by the paraboloid $r^2=4 z$

If Respond with true or false to the following assertion. Be prepared to defend your answer.

$S=\left\{(x, y, z): 1 \leq x^2+y^2+z^2 \leq 16\right\}$, then

$$\iiint_S d V=84 \pi$$

Evaluate $\iiint_S z^2 d V$, where S is the region bounded by $x^2+z=1$ and $y^2+z=1$ and the xy-plane.

In given problem, find the area of the indicated surface. Make a sketch in each case.

The part of the sphere $x^2+y^2+z^2=a^2$ inside the circular cylinder $x^2+y^2=a y(r=a \sin \theta$ in polar coordinates $)$, $a>0$

In given problem, find the moments of inertia $I_x, I_y$, and $I_z$ for the lamina bounded by the given curves and with the indicated density $\delta$.

$$y=\sqrt{x}, x=9, y=0 ; \delta(x, y)=x+y$$

In given problem, $R=\{(x, y): 0 \leq x \leq 6,0 \leq y \leq 4\}$ and P is the partition of R into six equal squares by the lines x=2, x=4, and y=2. Approximate $\iint_R f(x, y) d A$ by calculating the corresponding Riemann sum $\sum_{k=1}^6 f\left(\bar{x}_k, \bar{y}_k\right) \Delta A_k$, assuming that $\left(\bar{x}_k, \bar{y}_k\right)$ are the centers of the six squares.

$$f(x, y)=x^2+2 y^2$$

In given problem, $R=\{(x, y): 0 \leq x \leq 6,0 \leq y \leq 4\}$ and P is the partition of R into six equal squares by the lines x=2, x=4, and y=2. Approximate $\iint_R f(x, y) d A$ by calculating the corresponding Riemann sum $\sum_{k=1}^6 f\left(\bar{x}_k, \bar{y}_k\right) \Delta A_k$, assuming that $\left(\bar{x}_k, \bar{y}_k\right)$ are the centers of the six squares.

$$f(x, y)=10-y^2$$

In given problem, find the image of the rectangle with the given corners and find the Jacobian of the transformation.

$$x=u, y=u^2-v^2 ;(0,0),(3,0),(3,1),(0,1)$$