Approximation The base of a pile of sand at a cement plant is rectangular with approximate dimensions of $20$ meters by $30$ meters. If the base is placed on the $x y$-plane with one vertex at the origin, the coordinates on the surface of the pile are $(5,5,3),(15,5,6),(25,5,4),(5,15,2),(15,15,7)$, and $(25,15,3)$. Approximate the volume of sand in the pile.

Find the volume of the part of the ball $\rho \leqslant a$ that lies between the cones $\phi = \pi/6$ and $\phi = \pi/3$.

Find the volume of the given solid. Above the paraboloid

$$z = x^2 + y^2$$

and below the half-cone

$$z = √x^2+y^2$$

Find the volume of the given solid. The solid tetrahedron with vertices $(0,0,0),(0,0,1)$, $(0,2,0)$, and $(2,2,0)$

Use a computer algebra system to evaluate the iterated integral. ∫_0^a ∫_0^a-x (x² + y²) dy dx

Show that the function is a joint density function and find the required probability. f(x, y) = {e^-x-y, x ≥ 0, y ≥ 0 {0, elsewhere P(0 ≤ x ≤ 1, x ≤ y ≤ 1

Solve for a in the triple integral. ∫_0^1∫_0^3-a-y²∫_a^4-x-y² dz dx dy = 14/15

In this exercise, find the average value of the function over the given solid. The average value of a continuous function f(x, y, z) over a solid region Q is

$$\frac{1}{V} \iint_Q \int f(x, y, z) d V$$

where V is the volume of the solid region Q.

f(x, y, z)=x+y over the solid bounded by the sphere $x^2+y^2+z^2=2$

Show that the function is a joint density function and find the required probability. f(x, y) = {1/27(9 - x - y), 0 ≤ x ≤ 3, 3 ≤ y ≤ 6 {0, elsewhere P(0 ≤ x ≤ 1, 4 ≤ y ≤ 6)

Use a computer algebra system to evaluate the iterated integral. ∫_0^4∫_0^y 2 / (x + 1)(y + 1) dx dy

Find the solid region Q where the triple integral ∫∫_Q∫ (1 - 2x² - y² - 3z²) dV is a maximum. Use a computer algebra system to approximate the maximum value. What is the exact maximum value?

Find the mass and center of mass of the lamina for each density. R: triangle with vertices (0, 0), (0, a), (a, a) (a)ρ=k (b)ρ=ky (c)ρ=kx

Evaluate the integral.

$$∫_(e^y)^y (y ln x) / x dx, y > 0$$

In this exercise, find the average value of the function over the given solid. The average value of a continuous function f(x, y, z) over a solid region Q is

$$\frac{1}{V} \iint_Q \int f(x, y, z) d V$$

where V is the volume of the solid region Q.

f(x, y, z)=xyz over the cube in the first octant bounded by the coordinate planes, and the planes x=3, y=3, and z=3