Calculus II Mid-Trimester Quiz Buster

Calculate the mass of the plate shown in the following image if its density is given by:
𝜌(𝑥, 𝑦) = 3 + 2𝑥𝑦 (kg/m²).

The density function of a cylinder is given by:

𝜌(𝑟, 𝜃, 𝑧) = (3𝑟²)(2 + 0.75𝑠𝑖𝑛𝜃) (kg/m³)

The cylinder has a radius of 2m and a height of 3m. What is the mass of the cylinder?

An object is described as:

0 ≤ 𝑟 ≤ 2m

0 ≤ 𝛷 ≤ 2𝜋

0 ≤ 𝜃 ≤ π/2

Its density is given by: 𝜌(𝑟, 𝜃, 𝛷) = 6𝑟 (𝑘𝑔/𝑚³).

What is the total mass of this object?

A scalar field is given by 𝑓(𝑥, 𝑦) = 𝑥² + 𝑦². What is the gradient (∇𝑓(𝑥, 𝑦)) of this field?

What is the work done over the interval 𝑥 ∈ [0, 2] on the path 𝑦 = 𝑥² ? Here 𝑉̰ = (𝑥² + 1)𝑖̂ + (𝑥𝑦)𝑗̂.

Use Green’s Theorem to determine the line integral of a vector if the vector encloses a square with

points (0,0), (1,0), (1,1) and (0,1). 𝑉̰ = (6𝑥²𝑦)𝑖̂ + (2𝑦)𝑗̂.

Find the divergence of the vector field 𝐹̰ = (𝑥²ln(𝑦))𝑖̂ + (3𝑦sin(𝑧))𝑗̂ + (2cos(𝑥𝑦𝑧))𝑘̂

at the point (3, 1, 𝜋).

What is the net flow out of a cube with side lengths 1 unit if 𝑉̰ = 3𝑥𝑖̂ + 𝑦²𝑗̂ −(𝑧 + 6𝑥𝑦⁴)𝑘̂?

What is the curl of the vector field 𝑉̰ = 𝑥²𝑦 𝑖̂ − (𝑧³ − 3𝑥) 𝑗̂ + 4𝑦²𝑘̂?