In spherical coordinates, the inequalities 0≤ρ≤2, 3π/4 ≤ φ ≤ π 0 ≤ θ ≤ 2π, describe what kind of shape?

So, the radius is from 0 to 2, and it makes a full circle around the z-axis.
But what does 3π/4 ≤ φ ≤ π do to it?

It is like a 45 degree segment of a sphere with radius 2. Imagine an orange that can be broken into 8 segments. This would be one of them.

so would the shape be an cone ?

No. Did you read what I wrote previously?

It is a wedged-shaped segment of a sphere.

The inequality 3π/4 ≤ φ ≤ π limits the angle φ in spherical coordinates. To better understand how it affects the shape described by 0 ≤ ρ ≤ 2 and 0 ≤ θ ≤ 2π, let me explain the implications step by step.

First, let's consider the range of ρ, which is from 0 to 2. This means that the shape extends from the origin (ρ = 0) outward to a maximum distance of 2 units.

Next, we examine the range of θ, which is also from 0 to 2π. This indicates that the shape extends around the z-axis, forming a complete circular shape in the xy-plane.

Now, let's focus on the range of φ, which is 3π/4 ≤ φ ≤ π. In spherical coordinates, φ represents the angle formed with the positive z-axis. By having φ start at 3π/4 and extend to π, it restricts the shape to a specific portion of a sphere. More precisely, it describes the upper hemisphere (or the portion of a sphere above the xy-plane) within the given limits of the other coordinates.

Combining all the inequalities together, the shape described is a solid object, with a spherical boundary extending from the origin to a maximum distance of 2 units, forming a complete circular shape around the z-axis, and limited to the upper hemisphere above the xy-plane.

It's worth noting that the shape is more commonly referred to as a "spherical cap" rather than a full sphere, due to the upper hemisphere restriction imposed by φ.