Properties of Double Integrals

Calculus 3 · Axiom Academy

LESSON Properties of Double Integrals Six structural rules — linearity, additivity, sign, bounds, average, and the link to single integrals — that make tractable. The double integral is a linear operator : it passes straight through scalar multiples and sums. You can pull a constant out front, and split the integral of a sum into a sum of integrals — exactly as in single-variable calculus. If a region R splits into non-overlapping pieces R_1 and R_2 — sharing only a boundary curve — the integral over the whole is the sum of the integrals over the parts. This is the workhorse behind integrating over shapes that are not rectangles: cut the region into manageable pieces (above and below a curve, left and right of a line), integrate each, and add. The sign of follows the sign of f . Where the surface rises above the xy -plane it contributes positive volume; where it dips below, negative . Integration preserves order. If at every point of R , then their integrals obey the same inequality: Comparing f against the two constants m and M that bound it (so on R ) gives a way to sandwich the integral without ever evaluating it: The average value of f over R is the total integral divided by the area — the single constant height that would enclose exactly the same volume. Picture the average as the level a bumpy surface would settle to if you could pour it flat: the peaks fill the hollows, and the volume underneath is unchanged. 6. Connection to Single-Variable Integration

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