Questions from General Calculus

Q: Find the mass of a thin funnel in the shape of a

Find the mass of a thin funnel in the shape of a cone z = √x2 + y2, 1 < z < 4, if its density function is ρ (x, y, z) = 10 - z.

Q: (a). Give an integral expression for the moment of inertia

(a). Give an integral expression for the moment of inertia Iz about the z-axis of a thin sheet in the shape of a surface S if the density function is ρ. (b). Find the moment of inertia about the z-axi...

Q: Find the area of the surface. The part of the

Find the area of the surface. The part of the cone z = √x2 + y2 that lies between the plane y = x and the cylinder y = x2

Q: A fluid has density 870 kg/m3 and flows with velocity

A fluid has density 870 kg/m3 and flows with velocity v = z i + y2 j + x2 k, where x, y, and z are measured in meters and the components of v in meters per second. Find the rate of flow outward throug...

Q: Seawater has density 1025 kg/m3 and flows in a velocity

Seawater has density 1025 kg/m3 and flows in a velocity field v = y i + x j, where x, y, and z are measured in meters and the components of v in meters per second. Find the rate of flow outward throug...

Q: Use Gauss’s Law to find the charge contained in the solid hemisphere

Use Gauss’s Law to find the charge contained in the solid hemisphere x2 + y2 + z2 < a2, z > 0, if the electric field is E (x, y, z) = x i + y j + 2z k

Q: Let F (x, y, z) = (3x2yz

Let F (x, y, z) = (3x2yz - 3y) i + (x3z - 3x) j + (x3y + 2z) k Evaluate ∫C F ∙ dr, where C is the curve with initial point (0, 0, 2) and terminal point (0, 3, 0) s...

Q: Use Gauss’s Law to find the charge enclosed by the cube with

Use Gauss’s Law to find the charge enclosed by the cube with vertices (±1, ±1, ±1) if the electric field is E (x, y, z) = x i + y j + z k

Q: The temperature at the point (x, y, z)

The temperature at the point (x, y, z) in a substance with conductivity K = 6.5 is u (x, y, z) = 2y2 + 2z2. Find the rate of heat flow inward across the cylindrical surface y2 + z2 = 6, 0 < x < 4.

Q: The temperature at a point in a ball with conductivity K is

The temperature at a point in a ball with conductivity K is inversely proportional to the distance from the center of the ball. Find the rate of heat flow across a sphere S of radius a with center at...