PHYS2150 Chap.4 Double and Triple Integrals
Double and Triple Integrals
Regions come before antiderivatives
A double or triple integral accumulates a quantity over a region, so the first task is geometric. Draw the boundary curves or surfaces, locate their intersections, and choose a representative slice. For a vertical slice of a plane region, x advances through the outer interval while y travels from a lower boundary to an upper boundary. A horizontal slice reverses those roles.
Reversing the order of integration therefore requires a new description of the same set; simply exchanging the displayed limits usually changes the region.
Fubini turns regions into nested calculations
On a rectangle, a continuous function may be integrated in either order. On a general region, the same principle applies after the region is described as x-simple or y-simple.
If a typical slice enters or leaves through different curves in different parts of the projection, split the integral at the relevant intersection. This is not an algebraic inconvenience but an accurate record of what is counted.
A quick sample-point check inside each interval often detects reversed inner bounds before any integration begins.
Coordinate changes carry geometric scale
Polar coordinates are effective when circles, radial boundaries, or angular sectors dominate: x=r cosθ, y=r sinθ, and dA=r dr dθ. Cylindrical coordinates add z while retaining the same radial factor in dV=r dr dθ dz.
Under the course convention for spherical coordinates, x=ρ sinφ cosθ, y=ρ sinφ sinθ, z=ρ cosφ, and dV=ρ² sinφ dρ dφ dθ. The Jacobian factors are not optional; they express how coordinate rectangles stretch into physical area or volume.
General substitutions use a determinant
For x=g(u,v) and y=h(u,v), map the original boundaries into the uv-plane and multiply by the absolute Jacobian |∂(x,y)/∂(u,v)|.
The absolute value is needed because area remains nonnegative even if a transformation reverses orientation. Rewrite the integrand, the density, and any distance factors entirely in the new variables. Polar integration is the familiar special case whose Jacobian equals r.
A transformation is useful only if the simpler region or integrand compensates for this conversion work.
Physical integrals determine the integrand
Setting the integrand to one computes area or volume. A lamina with area density σ has mass M=∬σ dA; its centre of mass uses first moments divided by M. A solid with volume density ρ uses corresponding triple integrals.
Moment of inertia introduces squared perpendicular distance, such as x²+y² about the z-axis, and is nonnegative. Dimensional analysis distinguishes mass, first moment, and moment of inertia even when their limits are identical.
Audit coverage before evaluation
Check outer limits against the projection, inspect one typical inner interval, confirm angular coverage, and verify the Jacobian.
Symmetry reduces work only when both region and integrand possess the required symmetry. Some sphere–cone or sphere–cylinder intersections need multiple integrals because the active radial boundary changes with angle. Splitting at the equality of the competing bounds is the correct solution. Finally check scaling: with integrand one, doubling every length should multiply area by four and volume by eight.
What this chapter covers
- 01
Iterated integrals and slices
- 02
Order reversal
- 03
Polar area integrals
- 04
Cylindrical and spherical volumes
- 05
Jacobians and substitutions
- 06
Mass and centre of mass
- 07
Moments of inertia and symmetry
Worked example · free
Describe a polar crescent
- 2Solve boundary intersections.
- 3Order inner and outer radii.
- 2Include the area factor.
Key terms
- Iterated Integral
- A multiple integral evaluated through nested one-variable integrations.
- Polar Jacobian
- The radial factor r converting polar coordinate increments into area.
- Simple Region
- A region whose slices in a chosen direction are single intervals between two boundary functions.
- Change of Order
- A new slice-based description of the same region with the nesting of integrations reversed.
- Jacobian
- The determinant measuring local area or volume scaling under a change of variables.
- Moment of Inertia
- The integral of density times squared perpendicular distance from a specified axis.
- Centre of Mass
- The density-weighted mean position obtained by dividing first moments by total mass.
Double and Triple Integrals FAQ
Why might a solid need more than one integral?
The upper, lower or radial boundary can change across the projection. Splitting at the intersection curve faithfully describes the geometry, while forcing one set of limits can omit or double-count part of the solid. In spherical coordinates, compare competing radial endpoints and solve where they are equal; that angle is the natural place to divide the integral.
Can integration limits simply be swapped?
No. Reversing order means projecting the same region in a new direction and deriving new inner and outer bounds. The numerical symbols from the original order usually have different geometric roles. Sketch the region, choose a representative slice in the new direction, and split the projection if that slice changes its entry or exit curve.
How can a setup be checked without evaluation?
Plot or sketch the projection, inspect one representative inner interval, confirm coverage and the Jacobian, then check dimensions. Integrating one over a volume must scale with the cube of length. For mass or inertia, include density and distance factors in the unit check. A midpoint sample should satisfy every original boundary inequality.
Exam move
Build every solution in four passes. First sketch and label intersections. Second write the region in set notation before attaching an integrand. Third choose Cartesian, polar, cylindrical, spherical, or a general substitution and write the correct differential element from memory. Fourth audit coverage, units, symmetry, and sign before evaluating.
Practise the same plane region in both integration orders, because the comparison makes inner and outer limits concrete. For physical questions, write the element being accumulated—density times area or volume, signed distance for a first moment, squared perpendicular distance for inertia—before inserting bounds. Keep a short error log for missed Jacobians, incomplete angular ranges, and unsplit regions.
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