Find Volume Inside 3d Domains Polar Coordinates Calculator

Calculate Volume in 3D (Cylindrical Coordinates)

Enter the bounds for z, r, and theta (in degrees) to find the volume of the region.

2D projection of the integration region in the r-θ plane.

This page provides a Volume in Polar Coordinates Calculator (specifically using cylindrical coordinates with constant bounds for simplicity) and a detailed guide on calculating volumes of 3D domains described using these coordinate systems.

What is Volume in Polar/Cylindrical Coordinates?

When dealing with three-dimensional shapes that have some form of rotational symmetry or are conveniently described using angles and radii, cylindrical coordinates (r, θ, z) are often more suitable than Cartesian coordinates (x, y, z). Cylindrical coordinates extend polar coordinates (r, θ) by adding a z-axis perpendicular to the r-θ plane.

The volume of a region in cylindrical coordinates is found by evaluating the triple integral: V = ∫∫∫ r dz dr dθ, where 'r' is the Jacobian for the transformation from Cartesian to cylindrical coordinates. Our Volume in Polar Coordinates Calculator simplifies this for regions with constant boundaries for z, r, and θ.

This calculator is useful for students, engineers, and scientists who need to find the volume of shapes like cylinders, parts of cylinders, annular cylinders, or regions bounded by planes and cylinders where the base is described using polar coordinates.

Common Misconceptions

• Polar vs. Cylindrical: Polar coordinates are 2D (r, θ). When we talk about volume in "polar coordinates", we usually mean cylindrical (r, θ, z) or spherical (ρ, θ, φ) coordinates. This calculator uses cylindrical coordinates.
• Constant Bounds: This calculator assumes constant limits of integration for z, r, and θ. Real-world problems can involve variable limits (e.g., r depending on θ, or z depending on r and θ), requiring more complex integration.

Volume in Polar Coordinates Calculator Formula and Mathematical Explanation

The general formula for volume in cylindrical coordinates is:

V = ∫_θ1^θ2 ∫_r1(θ)^r2(θ) ∫_z1(r,θ)^z2(r,θ) r dz dr dθ

Where:

• z₁(r,θ) and z₂(r,θ) are the lower and upper surfaces bounding the volume.
• r₁(θ) and r₂(θ) are the inner and outer radial bounds, which can depend on θ.
• θ₁ and θ₂ are the start and end angles.
• The 'r' in 'r dz dr dθ' is the Jacobian determinant for the transformation from Cartesian to cylindrical coordinates.

Our Volume in Polar Coordinates Calculator uses a simplified version where z₁, z₂, r₁, and r₂ are constants:

V = ∫_θ1^θ2 ∫_r1^r2 ∫_z1^z2 r dz dr dθ

1. Integrate with respect to z: ∫_z1^z2 r dz = r[z]_z1^z2 = r(z₂ – z₁)

2. Integrate with respect to r: ∫_r1^r2 r(z₂ – z₁) dr = (z₂ – z₁)[r²/2]_r1^r2 = (z₂ – z₁) * (r₂² – r₁²)/2

3. Integrate with respect to θ: ∫_θ1^θ2 (z₂ – z₁) * (r₂² – r₁²)/2 dθ = (z₂ – z₁) * (r₂² – r₁²)/2 * [θ]_θ1^θ2 = (z₂ – z₁) * (r₂² – r₁²)/2 * (θ₂ – θ₁)

Here, θ₁ and θ₂ must be in radians for the final formula. The calculator takes degrees as input and converts them.

Variables Table

Variable	Meaning	Unit	Typical Range (for calculator)
z1, z2	Lower and upper z-bounds	Length units	Any real numbers
r1, r2	Inner and outer radii	Length units	r ≥ 0, r2 ≥ r1
θ1, θ2	Start and end angles	Degrees (input), Radians (calc)	0-360 degrees or more
V	Volume	Length units cubed	V ≥ 0

Practical Examples (Real-World Use Cases)

Example 1: Volume of a Full Cylinder

Find the volume of a cylinder with radius 3 and height 5.

• z₁ = 0, z₂ = 5 (height 5)
• r₁ = 0 (center), r₂ = 3 (radius 3)
• θ₁ = 0 degrees, θ₂ = 360 degrees (full circle)

Using the calculator or formula: V = (5-0) * 0.5 * (3² – 0²) * (2π – 0) = 5 * 0.5 * 9 * 2π = 45π ≈ 141.37 cubic units.

Example 2: Volume of an Annular Sector

Find the volume of a region between z=1 and z=3, between radii r=1 and r=2, and from θ=0 to θ=90 degrees (π/2 radians).

• z₁ = 1, z₂ = 3
• r₁ = 1, r₂ = 2
• θ₁ = 0 degrees, θ₂ = 90 degrees

V = (3-1) * 0.5 * (2² – 1²) * (π/2 – 0) = 2 * 0.5 * (4 – 1) * π/2 = 1 * 3 * π/2 = 3π/2 ≈ 4.71 cubic units.

How to Use This Volume in Polar Coordinates Calculator

Using the Volume in Polar Coordinates Calculator is straightforward:

1. Enter Z bounds: Input the lower (z₁) and upper (z₂) constant values for the z-coordinate.
2. Enter Radii: Input the inner (r₁) and outer (r₂) constant radii. Ensure r₁ ≥ 0 and r₂ ≥ r₁.
3. Enter Angles: Input the start (θ₁) and end (θ₂) angles in degrees. The calculator converts them to radians.
4. Calculate: The volume is calculated automatically. If not, click "Calculate Volume".
5. Read Results: The primary result is the calculated volume. Intermediate values like Δz, 0.5*(r₂²-r₁²), and Δθ (radians) are also shown.
6. Visualize: The SVG chart shows a top-down view of the region in the r-θ plane.
7. Reset: Click "Reset" to return to default values.

The calculator assumes constant bounds for z, r, and θ. For more complex shapes where bounds are functions, you would need to perform manual integration or use more advanced tools like a Triple Integrals Calculator.

Key Factors That Affect Volume Results

The calculated volume depends directly on the integration limits:

• Height (z₂ – z₁): The larger the difference between the upper and lower z bounds, the larger the volume.
• Radial Difference (r₂² – r₁²): The volume increases with the square of the radii. The area between r₁ and r₂ is crucial.
• Angular Sweep (θ₂ – θ₁): The larger the angle swept (in radians), the larger the volume, proportional to the angle. A full circle (360 degrees or 2π radians) gives the maximum volume for given r and z bounds.
• Inner Radius (r₁): A larger inner radius reduces the volume for a fixed outer radius.
• Outer Radius (r₂): A larger outer radius increases the volume.
• Units: Ensure all length units (for z and r) are consistent. The volume will be in cubic units of that length.

For more complex scenarios, the functions defining the bounds z(r, θ), r(θ) would be the key factors. Our Cylindrical Coordinates Volume calculator might handle some variations.

Frequently Asked Questions (FAQ)

What if the bounds are not constant?

If z, r, or θ bounds are functions (e.g., z=r, r=cos(θ)), you need to perform the integration ∫∫∫ r dz dr dθ step-by-step, substituting the functional bounds. This calculator is for constant bounds only. You might need a more general Integral Calculator.

What is the 'r' in 'r dz dr dθ'?

It's the Jacobian of the coordinate transformation from Cartesian (x, y, z) to cylindrical (r, θ, z). It accounts for the change in the differential volume element: dV = dx dy dz = r dr dθ dz.

Can I calculate the volume of a sphere using this?

Not directly with constant cylindrical bounds easily. A sphere is better described in spherical coordinates. You could do it by setting z bounds as functions of r, but it's complex. See our Spherical Coordinates Volume information.

Why are angles in degrees for input but radians for calculation?

Degrees are more intuitive for many users to input. However, mathematical formulas for integration involving angles almost always use radians. The calculator converts degrees to radians (180 degrees = π radians) before calculating.

What if r_inner is negative?

The radius 'r' in polar/cylindrical coordinates is defined as non-negative (r ≥ 0). The calculator will flag negative r_inner as invalid.

What if theta_end is smaller than theta_start?

The calculator will compute the difference theta_end – theta_start. If it's negative, the volume will be negative, which usually isn't physically meaningful unless direction is considered. You can add 360 degrees to theta_end if you mean to sweep across 0 degrees.

How does this relate to finding the area in polar coordinates?

Area in polar coordinates is A = ∫ 0.5 * r(θ)² dθ. This volume calculator essentially integrates the area of an annular sector (0.5*(r₂²-r₁²)dθ) over z. Check our Area in Polar Coordinates tool.

What are typical applications of this calculation?

Calculating volumes of pipes, tanks with cylindrical sections, volumes in engineering designs involving rotation, and in physics problems with cylindrical symmetry.

Related Tools and Internal Resources

• Cylindrical Coordinates Volume: For volumes specifically in cylindrical coordinates, possibly with more options.
• Spherical Coordinates Volume: For volumes better described using spherical coordinates (ρ, θ, φ).
• Triple Integrals Calculator: A more general tool for evaluating triple integrals with specified bounds.
• Area in Polar Coordinates: Calculate the area enclosed by a polar curve.
• Integral Calculator: For definite and indefinite integrals.
• Volume between Surfaces Calculator: Find the volume enclosed between two surfaces.