Triple Integrals
Integrate over a 3D region.
Key idea
Integrate over a 3D region.
In depth
∭ f dV sums over a 3D region, as an iterated integral. Volume corresponds to f = 1. Choose coordinates that fit the symmetry of the region.
Worked example
Volume of the box [0,1]×[0,2]×[0,3] is ∫₀³∫₀²∫₀¹ dx dy dz = 6.
Formula
Real-world uses
- Calculating the mass of a 3D object with varying density, such as a rock or a machine part
Common mistake
Getting limits wrong for non-rectangular regions.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Triple Integrals?
Integrate over a 3D region. It is a topic at the expert level in Calculus.
What is the formula for Triple Integrals?
The key formula is ∭ f dV.
Where is Triple Integrals used in real life?
Real-world examples: Calculating the mass of a 3D object with varying density, such as a rock or a machine part.
What should I learn before Triple Integrals?
Make sure you are comfortable with Double Integrals in Polar, the previous topic in Calculus.
What is a common mistake with Triple Integrals?
Getting limits wrong for non-rectangular regions.
How can I practise Triple Integrals?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Triple Integrals?
Move on to Cylindrical and Spherical next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.