IntermediateTopic 43 of 104Engineering Mathematics
Multiple integrals
Double and triple integrals compute areas, volumes, mass and inertia over regions.
In depth
Double integrals ∬f dA sum over areas and triple integrals over volumes, evaluated as iterated integrals with suitable limits. They compute mass, moments of inertia, charge and probabilities. Changing the order of integration may simplify the work.
Key points
- Jacobian for change of variables
- Polar coordinates for circular regions
- Sketch the region first
ExampleThe area of a unit circle is ∬ r dr dθ = π.
Double and triple integrals find areas, masses and inertias of plates and solids, which are needed for accurate structural and vehicle calculations.
What is Multiple integrals?
Double and triple integrals compute areas, volumes, mass and inertia over regions.
Why is it important to understand multiple integrals?
Double and triple integrals find areas, masses and inertias of plates and solids, which are needed for accurate structural and vehicle calculations.
When should I switch the order of integration?
Switch when the region or integrand becomes simpler in the other order, which is often the case for triangular regions.
Why use polar coordinates for a double integral?
Circular regions and radially symmetric integrands are much simpler in polar form, and the area element becomes r dr d theta.
How is Multiple integrals used in practice?
Typical applications include computing the mass of a plate with varying density, finding the moment of inertia of a wheel for a dynamics model and calculating the volume of a non-rectangular reservoir.
What should I learn before Multiple integrals?
Study "Gradient" first (topic 42 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Multiple integrals?
Jacobian for change of variables. Polar coordinates for circular regions. Sketch the region first.
Can you give an example of Multiple integrals?
The area of a unit circle is ∬ r dr dθ = π.
What comes after Multiple integrals?
The next topic is "Coordinate systems", which builds directly on this one.
How difficult is Multiple integrals?
It is classed as Intermediate in Engineering Mathematics (topic 43 of 104). It is an intermediate topic. Work through the beginner topics in Engineering Mathematics first.