spherical coordinates for hemisphere

Definition of spherical coordinates ρ = distance to origin, ρ ≥ 0 φ = angle to z-axis, 0 ≤ φ ≤ π θ = usual θ = angle of projection to xy-plane with x-axis, 0 ≤ θ ≤ 2π Easy trigonometry gives: z = ρcosφ x = ρsinφcosθ y = ρsinφsinθ

{(r, θ, φ) : secφ ≤ r ≤ 2 cosφ, 0 ≤ φ ≤ π 4 , 0 ≤ θ < 2π} describes the hemisphere centered at (0, 0, 1) with radius 1 unit

Find the z coordinate of the center of mass of the solid consisting of the part of the hemisphere z = √4 − x2 − y2 inside the cylinder x2 + y2 = 2x if the density ρ = 

and spherical coordinates (r, θ, φ) are the topic of this and the next sections The equation of the upper hemisphere in cylindrical coordinates is r = √ a2 − z2

Set up a triple integral in cylindrical coordinates representing the volume of the bead Evaluate the integral Solution In cylindrical coordinates, the sphere is given 

3 Find the volume and the center of mass of a diamond, the intersection of the unit sphere with the cone given in cylindrical coordinates as z = √3r Solution: we 

Use spherical coordinates to express region between the sphere x2 + y2 + z2 = 1 and the cone z = √ x2 + y2 Solution: (x = ρsin(φ) cos(θ), y = ρsin(φ) sin(θ), 

1P1 Calculus 2 Example: By transforming to spherical polar coordinates, integrate the function ( )2/32 2 2 z y xf + + = over the hemisphere defined by 9 2 2

cylindrical and spherical coordinates are also illustrated I hope this helps The equation for the outer edge of a sphere of radius a is given by x2 + y2 + z2 = a2

25 oct 2019 · cylinder, cone, sphere, we can often simplify our work by using cylindrical or spherical coordinates, which are introduced in the lecture