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

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

Let the plane region R be a unit circle and let the maximum value of f on R be 6. Is the greatest possible value of $\int_R \int f(x, y) d y d x$ equal to 6? Why or why not? If not, what is the greatest possible value?

A joint density function of the continuous random variables x and y is a function f(x, y) satisfying the following property.

$f(x, y) \geq 0$ for all (x, y)

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 $x^2+y^2=9, \quad z \geq 0$

Evaluate the iterated integral. ∫_0^3 ∫_0^π/3 ∫_0^4 r cos θ dr dθ dz

Evaluate the iterated integral. ∫_0^π/2 ∫_0^π/2 ∫_0^2 ρ² dp dθ dΦ

In the following exercise, evaluate the triple iterated integral.

$$\int_0^{\pi / 2} \int_0^\pi \int_0^2 e^{-\rho^3} \rho^2 d \rho d \theta d \phi$$

Find the mass and the indicated coordinates of the center of mass of the solid region Q of density ρ bounded by the graphs of the equations. Find y̅ using ρ(x, y, z) = kx. Q: z = 5 - y, z = 0, y = 0, x = 0, x = 5

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.∫_0^9∫_√x^3 dy dx

Evaluate the integral. $\int_{x^{3}}^{\sqrt{x}}\left(x^{2}+3 y^{2}\right) d y$

In the following exercise, evaluate the integral.

$$\int_{x^3}^{\sqrt{x}}\left(x^2+3 y^2\right) d y$$

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)

Set up a triple integral that gives the moment of inertia about the z-axis of the solid region Q of density ρ. Q = {(x, y, z): -1 ≤ x ≤ 1, -1 ≤ y ≤ 1, 0 ≤ z ≤ 1-x}, ρ = √x²+y²+z²

The table shows values of a function f over a square region R. Divide the region into 16 equal squares and select $\left(x_i, y_i\right)$ to be the point in the ith square closest to the origin. Approximate the value of the integral below. Compare this approximation with that obtained by using the point in the ith square farthest from the origin.

$$\int_0^4 \int_0^4 f(x, y) d y d x$$

Find the area of the surface given by $z=$ $f(x, y)$ over the region $R$. (Hint: Some of the integrals are simpler in polar coordinates.) \

$$f(x, y)=9-x^2$$

$R$ : square with vertices $(0,0),(3,0),(0,3),(3,3)$