Let W be part of the solid sphere of radius 4, centered at the origin, that lies above the plane z = 2. Set up an integral in cylindrical coordinates to find the volume of the region. Do NOT actually integrate.

V = ∫∫∫rdrdθdz integrating from z = 2 to z = 4, r = 0 to √(16 - z²) and θ = 0 to 2π

Step-by-step explanation:

Since we have the radius of the sphere R = 4, we have R² = r² + z² where r = radius of cylinder in z-plane and z = height² of cylinder.

So, r = √(R² - z²)

r = √(4² - z²)

r = √(16 - z²)

Since the region is above the plane z = 2, we integrate z from z = 2 to z = R = 4

Our volume integral in cylindrical coordinates is thus

V = ∫∫∫rdrdθdz integrating from z = 2 to z = 4, r = 0 to √(16 - z²) and θ = 0 to 2π

chemists are often responsible for designing a chemical reaction and analyzing the products obtained from it. one significant part of designing a chemical reaction is determining the amount of each reactant that is needed and the amount of each product that will be produced. in order to do so, chemists use stoichiometry, the quantitative relationships between reactants and products.