Questions from General Calculus

Q: (a). Show that ∫_0^1 ∫_0^

(a). Show that ∫_0^1 ∫_0^1 ∫_0^1 1/(1-xyz) dx dy dz = ∑_(n-1)^∞ 1/n^3 (Nobody has ever been able to find the exact value of the sum of this series.) (b). Show that ∫_0^1 ∫_0^1 ∫_0^1 1/(1-xyz) dx dy d...

Q: Assume that the solid has constant density k. Find the

Assume that the solid has constant density k. Find the moment of inertia about the z-axis of the solid cylinder x2 + y2 < a2, 0 < z < h.

Q: Assume that the solid has constant density k. Find the

Assume that the solid has constant density k. Find the moment of inertia about the z-axis of the solid cone √(x^2 +y^2 ) < z < h.

Q: The surfaces ρ = 1 + 1/5 sin mθ sin

The surfaces ρ = 1 + 1/5 sin mθ sin nφ have been used as models for tumors. The “bumpy sphere” with m = 6 and n = 5 is shown. Use...

Q: Show that ∫_(-∞)^∞ ∫_(-∞)^∞ ∫_(-∞)^∞√(x^2+y^2

Show that ∫_(-∞)^∞ ∫_(-∞)^∞ ∫_(-∞)^∞√(x^2+y^2+y^2 ) e^(-(x^2+y^2+z^2)) dx dy dz = 2π (The improper triple integral is defined as the limit of a triple integral over a solid sphere as the radius of th...

Q: Let E be the solid in the first octant bounded by the

Let E be the solid in the first octant bounded by the cylinder x2 + y2 = 1 and the planes y = z, x = 0, and z = 0 with the density function ρ (x, y, z) = 1 + x + y + z. Use a computer algebra system...

Q: If E is the solid of Exercise 18 with density function ρ

If E is the solid of Exercise 18 with density function ρ (x, y, z) = x2 + y2, find the following quantities, correct to three decimal places. Exercise 18: Evaluate the triple integral. ∭E z dV, wher...

Q: Evaluate the integral by reversing the order of integration. ∫_

Evaluate the integral by reversing the order of integration. ∫_0^1 ∫_3y^3e^(x^2 ) dx dy

Q: Suppose X, Y, and Z are random variables with joint

Suppose X, Y, and Z are random variables with joint density function f (x, y, z) = Ce-(0.5x+0.2y+0.1z) if x > 0, y > 0, z > 0, and f (x, y, z) = 0 otherwise. (a). Find the value of the constant C. (b)...

Q: The average value of a function f (x, y,

The average value of a function f (x, y, z) over a solid region E is defined to be where V (E) is the volume of E. For instance, if is a density function, then ρ_(¬ave) is the...