A triple integral adds up tiny pieces of a three-dimensional region. Instead of summing thin strips or small rectangles, it sums tiny volume elements dV throughout a solid. This makes triple integrals a main tool for finding volumes, masses, centers of mass, and total amounts in space.
They matter because many real objects and fields vary in all three directions.
Key Facts
- Volume of a solid region R: V = ∭_R 1 dV
- Mass with density ρ(x, y, z): m = ∭_R ρ(x, y, z) dV
- Rectangular coordinates: dV = dx dy dz, with the order chosen to match the limits
- For a box a ≤ x ≤ b, c ≤ y ≤ d, e ≤ z ≤ f: ∭_R f(x, y, z) dV = ∫_a^b ∫_c^d ∫_e^f f(x, y, z) dz dy dx
- The inner integral limits may depend on the outer variables, such as z from g1(x, y) to g2(x, y)
- Cylindrical coordinates use dV = r dz dr dθ, and spherical coordinates use dV = ρ^2 sinφ dρ dφ dθ
Vocabulary
- Triple integral
- A triple integral is an integral that sums a quantity over a three-dimensional region.
- Volume element
- A volume element dV is a tiny piece of space used as the basic unit being added in a triple integral.
- Iterated integral
- An iterated integral evaluates a multiple integral one variable at a time in a chosen order.
- Density function
- A density function ρ(x, y, z) gives the amount of mass per unit volume at each point in space.
- Limits of integration
- Limits of integration describe the boundaries of the region being integrated over.
Common Mistakes to Avoid
- Using the wrong order of limits: The innermost limits must match the innermost differential, so ∫∫∫ f dz dy dx means z is integrated first.
- Forgetting the volume element factor in new coordinates: In cylindrical coordinates dV is r dz dr dθ, not just dz dr dθ.
- Treating variable limits as constants: If z runs from 0 to 4 - x - y, the upper limit changes with x and y and cannot be replaced by one number.
- Integrating before understanding the region: A sketch or projection is often needed because the limits describe geometry, not just algebra.
Practice Questions
- 1 Find the volume of the rectangular box 0 ≤ x ≤ 2, 0 ≤ y ≤ 3, 0 ≤ z ≤ 4 using a triple integral.
- 2 A solid region R is given by 0 ≤ x ≤ 1, 0 ≤ y ≤ 2, 0 ≤ z ≤ x + y. Set up and evaluate ∭_R 1 dV.
- 3 Explain why cylindrical coordinates are usually a better choice than rectangular coordinates for integrating over a solid cylinder.