Determine which value best approximates the volume of the solid between the $x y$-plane and the function over the region. (Make your selection on the basis of a sketch of the solid and not by performing any calculations.)
$f(x, y)=15-2 y ; R$ : semicircle: $x^2+y^2=16, y \geq 0$
(a) $100$ (b) $200$ (c ) $300$ (d) $-200$ (e) $800$

Find the moment of inertia $I_z$ of the solid of indicated density.
The solid of uniform density inside the paraboloid $z=16-x^2-y^2$ and outside the cylinder $x^2+y^2=9, \quad z \geq 0$

Find the average value of f(x, y) over the plane region R. f(x, y) = sin(x + y) R: rectangle with vertices (0, 0), (π, 0), (π, π), (0, π)

Verify the moments of inertia for the solid of uniform density. Use a graphing utility to evaluate the triple integrals.

$$\begin{aligned} & I_x=\frac{1}{12} m\left(a^2+b^2\right) \\ & I_y=\frac{1}{12} m\left(b^2+c^2\right) \\ & I_z=\frac{1}{12} m\left(a^2+c^2\right)\end{aligned}$$

In the following exercise, use a graphing utility to approximate the iterated integral.

$$\int_0^{\pi / 4} \int_0^4 5 r e^{\sqrt{r \theta}} d r d \theta$$

Sketch the region R whose area is given by the iterated integral. Then switch the order of integration and show that both orders yield the same area.

$$∫_0^4∫_0^x/2 dy dx + ∫_4^6∫_0^6-x dy dx$$

Find the center of mass of the solid of uniform density bounded by the graphs of the equations.
$x^2+y^2+z^2=25, z=4$ (the larger solid)

Find the average value of f(x, y) over the plane region R. f(x, y) = x R: rectangle with vertices (0, 0), (4, 0), (4, 2), (0, 2)

Use a computer algebra system to approximate the iterated integral. ∫_π/4^π/2∫_0^5 r√1+r³ sin √θ dr dθ

Find the average value of f(x, y) over the plane region R. f(x, y) = 1 / x+y, R: triangle with vertices (0, 0), (1, 0), (1, 1)

Find the center of mass of the solid of uniform density bounded by the graphs of the equations.
$x^2+y^2+z^2=a^2$, first octant

Horizontal cross sections of a piece of ice that broke from a glacier are in the shape of a quarter of a circle with a radius of approximately 50 feet. The base is divided into 20 subregions, as shown in the figure. At the center of each subregion, the height of the ice is measured, yielding the following points in cylindrical coordinates.

$\left(5, \frac{\pi}{16}, 7\right)$,$\left(15, \frac{\pi}{16}, 8\right)$,$\left(25, \frac{\pi}{16}, 10\right)$,$\left(35, \frac{\pi}{16}, 12\right)$,$\left(45, \frac{\pi}{16}, 9\right)$, $\left(5, \frac{3 \pi}{16}, 9\right)$, $\left(15, \frac{3 \pi}{16}, 10\right)$, $\left(25, \frac{3 \pi}{16}, 14\right)$, $\left(35, \frac{3 \pi}{16}, 15\right)$, $\left(45, \frac{3 \pi}{16}, 10\right)$, $\left(5, \frac{5 \pi}{16}, 9\right)$, $\left(15, \frac{5 \pi}{16}, 11\right)$, $\left(25, \frac{5 \pi}{16}, 15\right)$, $\left(35, \frac{5 \pi}{16}, 18\right)$, $\left(45, \frac{5 \pi}{16}, 14\right)$, $\left(5, \frac{7 \pi}{16}, 5\right)$, $\left(15, \frac{7 \pi}{16}, 8\right)$, $\left(25, \frac{7 \pi}{16}, 11\right)$, $\left(35, \frac{7 \pi}{16}, 16\right)$, $\left(45, \frac{7 \pi}{16}, 12\right)$

There are 7.48 gallons of water per cubic foot. Approximate the number of gallons of water in the piece of ice.

Verify the moments of inertia for the solid of uniform density. Use a graphing utility to evaluate the triple integrals.

$$\begin{aligned} & I_x=\frac{1}{12} m\left(3 a^2+L^2\right) \\ & I_y=\frac{1}{2} m a^2 \\ & I_z=\frac{1}{12} m\left(3 a^2+L^2\right) \end{aligned}$$

Horizontal cross sections of a piece of ice that broke from a glacier are in the shape of a quarter of a circle with a radius of approximately 50 feet. The base is divided into 20 subregions, as shown in the figure. At the center of each subregion, the height of the ice is measured, yielding the following points in cylindrical coordinates.

$\left(5, \frac{\pi}{16}, 7\right)$,$\left(15, \frac{\pi}{16}, 8\right)$,$\left(25, \frac{\pi}{16}, 10\right)$,$\left(35, \frac{\pi}{16}, 12\right)$,$\left(45, \frac{\pi}{16}, 9\right)$, $\left(5, \frac{3 \pi}{16}, 9\right)$, $\left(15, \frac{3 \pi}{16}, 10\right)$, $\left(25, \frac{3 \pi}{16}, 14\right)$, $\left(35, \frac{3 \pi}{16}, 15\right)$, $\left(45, \frac{3 \pi}{16}, 10\right)$, $\left(5, \frac{5 \pi}{16}, 9\right)$, $\left(15, \frac{5 \pi}{16}, 11\right)$, $\left(25, \frac{5 \pi}{16}, 15\right)$, $\left(35, \frac{5 \pi}{16}, 18\right)$, $\left(45, \frac{5 \pi}{16}, 14\right)$, $\left(5, \frac{7 \pi}{16}, 5\right)$, $\left(15, \frac{7 \pi}{16}, 8\right)$, $\left(25, \frac{7 \pi}{16}, 11\right)$, $\left(35, \frac{7 \pi}{16}, 16\right)$, $\left(45, \frac{7 \pi}{16}, 12\right)$

Ice weighs approximately 57 pounds per cubic foot. Approximate the weight of the piece of ice.