# Calculus III: Sketching Solids with Spherical Coordinates

• acid_rain
In summary, the solid described in this conversation consists of all points with spherical coordinates (ρ, θ, φ) where 0 ≤ θ ≤ π/2, 0 ≤ φ ≤ π/6, and 0 ≤ ρ ≤ 2cosφ. The range of ρ is dependent on the value of φ, with a maximum value of √3 at φ = 0 and a minimum value of 0 at φ = π/6. The volume can be completed by varying θ.
acid_rain
I really have difficulty with spherical coordinates graphings. can someone help me with this problem?

Sketch the solid consisting of all points with spherical coordinates (ρ,θ,φ) such that 0≤ θ ≤π/2, 0≤ φ ≤π/6, and 0≤ ρ ≤2cosφ

thanks so much!

i think this looks like a bowl with radius ρ= root 3,
then I don't understand how to graph theta and phi according to that range. that's where i got stuck.

Last edited:
The "range" of ρ is dependant upon the value of φ. So if you start at φ=0. At this point, all the ρ btw 0 and 2 are permitted. But as φ progresses towards π/6, ρ can ony take value btw 0 and 2cosφ. And at φ=π/6, it can only take values btw 0 and $\sqrt{3}$. In btw, the range of rho decrease as cosφ decreases btw 0 and π/6, i.e. very slowly at the beginning (the crest of the cos) and then steadily until π/6.

Now that you've got the range of ρ and φ covered, just complete the volume by making θ vary.

## 1. What is the purpose of using spherical coordinates in sketching solids?

The purpose of using spherical coordinates in sketching solids is to represent three-dimensional objects in a way that is more intuitive and efficient than using rectangular coordinates. Spherical coordinates use a radial distance, an angle from the positive z-axis, and an angle from the positive x-axis to locate points in three-dimensional space, making it easier to visualize and understand the shape of the solid.

## 2. How are spherical coordinates related to polar coordinates?

Spherical coordinates are similar to polar coordinates in that they use an angle and a distance to locate points. However, polar coordinates are used in two dimensions, while spherical coordinates are used in three dimensions. The angle in spherical coordinates represents the angle from the positive z-axis instead of the positive x-axis as in polar coordinates.

## 3. Can any solid be represented using spherical coordinates?

Yes, any solid can be represented using spherical coordinates. However, some solids may be more complex to sketch using spherical coordinates, and it may be more efficient to use other coordinate systems such as cylindrical or rectangular coordinates.

## 4. How do you convert between spherical and rectangular coordinates?

To convert from spherical coordinates (ρ, θ, φ) to rectangular coordinates (x, y, z), you can use the following equations:

x = ρsinφcosθ

y = ρsinφsinθ

z = ρcosφ

To convert from rectangular coordinates (x, y, z) to spherical coordinates (ρ, θ, φ), you can use the following equations:

ρ = √(x² + y² + z²)

θ = arctan(y/x)

φ = arccos(z/√(x² + y² + z²))

## 5. How can you use spherical coordinates to calculate volume and surface area of solids?

Spherical coordinates can be used to calculate volume and surface area of solids by using triple integrals. The volume of a solid can be calculated by integrating the function over the region of the solid in spherical coordinates. The surface area of a solid can be calculated by integrating the function representing the surface over the solid's region in spherical coordinates. This method is particularly useful for calculating volumes and surface areas of solids with spherical symmetry, such as spheres or cones.

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