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163 LearnersLast updated on September 3, 2025
Volume of Ellipsoid Using Triple Integration
The volume of an ellipsoid can be determined using triple integration, which provides a method to calculate the total space it occupies. An ellipsoid is a 3D shape resembling a stretched or compressed sphere, defined by three semi-axes. To find the volume of an ellipsoid, we use integration over its defining region in three-dimensional space. This topic will explore how to compute the volume of an ellipsoid using triple integration.
What is the volume of an ellipsoid?
The volume of an ellipsoid is the amount of space it occupies. It can be calculated using the triple integral over the ellipsoid's region.
The standard formula is: Volume = (4/3)πabc Where ‘a,’ ‘b,’ and ‘c’ are the semi-axes of the ellipsoid.
Volume of Ellipsoid Formula An ellipsoid is a 3-dimensional shape with three orthogonal axes. To calculate its volume, you can apply triple integration over the region defined by the ellipsoid.
Alternatively, the formula for the volume of an ellipsoid is: Volume = (4/3)πabc
How to Derive the Volume of an Ellipsoid?
To derive the volume of an ellipsoid, we consider the integral over the region it occupies.
The ellipsoid is typically defined by: (x/a)² + (y/b)² + (z/c)² ≤ 1
Using spherical coordinates and appropriate scaling, the volume can be derived through integration: Volume = ∫∫∫_V dV
This results in the formula: Volume = (4/3)πabc
How to find the volume of an ellipsoid using triple integration?
The volume of an ellipsoid is expressed in cubic units, such as cubic centimeters (cm³) or cubic meters (m³).
To find the volume using triple integration, set up the integral over the ellipsoid's region: Volume = ∫∫∫_V dV
Transform into spherical coordinates: Volume = ∫(from 0 to 2π) ∫(from 0 to π) ∫(from 0 to 1) abc sin(θ) dρ dθ dφ
Evaluate the integral to obtain the volume: Volume = (4/3)πabc
Tips and Tricks for Calculating the Volume of an Ellipsoid
Remember the formula: The formula for the volume of an ellipsoid is straightforward: Volume = (4/3)πabc
Break it down: The volume is the space inside the ellipsoid. You need the lengths of the three semi-axes (a, b, c).
Simplify the numbers: If the semi-axes are simple numbers like 2, 3, or 4, multiplying them is straightforward. For example, a=2, b=3, c=4 gives (4/3)π(2)(3)(4).
Check your setup: Ensure the region of integration correctly represents the ellipsoid.
Common Mistakes and How to Avoid Them in Volume of Ellipsoid
Mistakes are common when learning to calculate the volume of an ellipsoid. Let’s explore some common errors and how to avoid them for a better understanding.
Mistake 1
Confusing Volume with Surface Area
Some students confuse the formula for volume with the formula for surface area. Surface area involves more complex expressions, but volume is calculated using the ellipsoid formula: (4/3)πabc.
Mistake 2
Incorrectly Setting Up the Integral
Errors occur when setting up the triple integral. Ensure the limits and the function inside the integral match the ellipsoid's equation.
Mistake 3
Using the Wrong Formula for a Sphere
Some use the formula for the volume of a sphere (4/3)πr³ instead of the ellipsoid formula (4/3)πabc. Remember that an ellipsoid has three distinct axes.
Mistake 4
Ignoring the Limits of Integration
Ensure the integration limits correctly reflect the ellipsoid's boundaries. Incorrect limits lead to errors in the computed volume.
Mistake 5
Neglecting to Scale for Ellipsoids
When transforming to spherical coordinates, remember to scale by the semi-axes a, b, and c. Failing to do so results in incorrect volume calculations.
Volume of Ellipsoid Using Triple Integration Examples
Problem 1
An ellipsoid has semi-axes of 3 cm, 4 cm, and 5 cm. What is its volume?
The volume of the ellipsoid is 251.33 cm³.
Explanation
To find the volume of an ellipsoid, use the formula: V = (4/3)πabc
Here, a=3 cm, b=4 cm, c=5 cm,
so: V = (4/3)π(3)(4)(5) = 251.33 cm³
Problem 2
An ellipsoid has semi-axes 2 m, 2 m, and 3 m. Find its volume.
The volume of the ellipsoid is 50.27 m³.
Explanation
To find the volume of an ellipsoid, use the formula: V = (4/3)πabc
Substitute a=2 m, b=2 m, c=3 m:
V = (4/3)π(2)(2)(3) = 50.27 m³
Problem 3
The volume of an ellipsoid is 400 cm³. If two semi-axes are 5 cm and 8 cm, what is the third semi-axis?
The third semi-axis of the ellipsoid is approximately 3.82 cm.
Explanation
Given the volume and two semi-axes, solve for the third:
V = (4/3)πabc 400 = (4/3)π(5)(8)c
c ≈ 3.82 cm
Problem 4
An ellipsoid has semi-axes of 1.5 inches, 2 inches, and 3 inches. Find its volume.
The volume of the ellipsoid is 37.70 inches³.
Explanation
Using the formula for volume: V = (4/3)πabc
Substitute a=1.5 inches, b=2 inches, c=3 inches:
V = (4/3)π(1.5)(2)(3) = 37.70 inches³
Problem 5
You have an ellipsoid with semi-axes of 1 foot, 1 foot, and 2 feet. How much space (in cubic feet) does it occupy?
The ellipsoid occupies 8.38 cubic feet.
Explanation
Using the formula for volume: V = (4/3)πabc
Substitute a=1 foot, b=1 foot, c=2 feet:
V = (4/3)π(1)(1)(2) = 8.38 ft³
FAQs on Volume of Ellipsoid Using Triple Integration
1.Is the volume of an ellipsoid the same as the surface area?
2.How do you find the volume if the semi-axes are given?
3.What if I have the volume and need to find a semi-axis?
4.Can the semi-axes be decimals or fractions?
5.Is the volume of an ellipsoid the same as the surface area?
Important Glossaries for Volume of Ellipsoid Using Triple Integration
- Ellipsoid: A 3-dimensional shape defined by three semi-axes, forming a stretched or compressed sphere.
- Semi-axes: The three principal radii (a, b, c) of an ellipsoid, determining its dimensions.
- Volume: The amount of space enclosed within a 3D object, expressed in cubic units (e.g., cm³, m³).
- Triple Integration: A method of integrating a function over a three-dimensional region, used to find volumes.
- Spherical Coordinates: A coordinate system used to simplify integration in spherical or ellipsoidal regions.
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Seyed Ali Fathima S
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Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.
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