Multiple Integrals
“A multiple integral is 'multiply each little patch of area by its height, then add it all up' — a volume.”
The formula
∬_R f(x, y) dA = ∫∫ f(x, y) dx dyHow to read it: Over a region R, integrate f once in x and once in y to find the volume trapped beneath the surface
- ∬_R
- — The double-integral sign over a region R
- dA
- — Area element dA = one tiny patch of area
- dx dy
- — Inner integral (x) first, outer integral (y) after — an iterated integral
The hook
If a 1-D integral is a ruler for area, a 2-D integral is a ruler for volume — it captures the whole space between a surface and the floor.
In plain words
You chop the base region R into tiny rectangles, multiply each patch's area by the function's height above it to make a little column, and add up every column.
The intuition
Imagine a city map where building height is f(x,y). For each block (a small area dA) compute 'area × height = that block's volume', then sum over the whole city for total volume. There are two integral signs because you sweep once across x and once across y.
How it's built
An iterated integral is computed inside-out: first integrate in x with y held fixed (giving the area of one cross-sectional slice), then integrate that in y (stacking the slices into a volume). Holding y constant in the inner integral mirrors the partial derivative exactly.
Example
Compute ∫₀¹∫₀¹ (x+y) dx dy. Inner (x): ∫₀¹(x+y)dx = [x²/2 + yx]₀¹ = 1/2 + y. Outer (y): ∫₀¹(1/2 + y)dy = [y/2 + y²/2]₀¹ = 1/2 + 1/2 = 1. The volume is 1.
Common mistake
'Two integral signs means twice as hard' is wrong — you do one at a time, treating the other variable as a constant, in sequence. For a smooth region it makes no difference whether you integrate x first or y first (Fubini's theorem).
Where it's used
Mass, center of mass, and moment of inertia (integrating density over a region), joint probability distributions (total 2-variable probability = 1), electromagnetic flux, and lighting integrals in computer graphics — any 'total of a quantity spread over a region' is a multiple integral.
Where it came from
An extension of Newton and Leibniz's integral to several variables; in the 19th century Fubini proved that you may rewrite it as iterated single integrals, opening the door to practical computation.
Prerequisites
Quick check
What is ∫₀¹∫₀¹ 1 dx dy? (volume of height 1 over the unit square)
- 0
- 1✓
- 2
- 1/2
Practice
Compute ∫₀²∫₀³ 1 dx dy. (volume of a box of height 1 over a 3×2 base)
Answer: 6
- Inner: ∫₀³ 1 dx = 3
- Outer: ∫₀² 3 dy = 3·2 = 6
Takeaway: Integrating the constant 1 gives area of region × height = volume.
Compute ∫₀¹∫₀¹ (x+y) dx dy.
Answer: 1
- Inner (x): ∫₀¹(x+y)dx = 1/2 + y
- Outer (y): ∫₀¹(1/2 + y)dy = 1/2 + 1/2 = 1
Takeaway: Iterated integrals go inside-out — y is constant in the inner one.
Compute ∫₀²∫₀² xy dx dy.
Answer: 4
- Inner (x): ∫₀² xy dx = y·[x²/2]₀² = 2y
- Outer (y): ∫₀² 2y dy = [y²]₀² = 4
Takeaway: When variables separate you may factor: (∫x dx)(∫y dy) = 2·2 = 4.
Explain in your notebook why swapping the order (x first vs y first) gives the same answer (Fubini's theorem).
Answer: undefined
- The volume is one real object — stacking vertical slices or horizontal slices fills the same space
- For a continuous (integrable) function on a bounded region, both iterated integrals equal that volume
- This is Fubini's theorem — the condition under which order may be swapped
Takeaway: For smooth functions the order is free — there is only one underlying volume.