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

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.

Translate the lamina R: square with vertices (0, 0), (a, 0), (0, a), (a, a) to the right five units and determine the resulting center of mass.

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.

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

Approximate the volume of the piece of ice.

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

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. ∫_0^2∫_x/2^1 dy dx

Sketch the solid whose volume is given by the iterated integral and rewrite the integral using the indicated order of integration. ∫_0^6∫_0^6-x∫_0^6-x-y dz dy dx Rewrite using the order dy dx dz.

In the following exercise, verify the moments of inertia for the solid of uniform density. Use a graphing utility to evaluate the triple integrals.

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

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, evaluate the triple iterated integral.

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

Use an iterated integral to find the area of the region bounded by the graphs of the equations. y = 6x - x², y = x² - 2x

Approximate the triple iterated integral. ∫_0^3∫_-√9-y²^√9-y²∫_0^y² y dz dx dy

The temperature in degrees Celsius on the surface of a metal plate is T(x, y) = 20 - 4x² - y² where x and y are measured in centimeters. Estimate the average temperature when x varies between 0 and 2 centimeters and y varies between 0 and 4 centimeters.