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Calculus · Quick Reference
Multiple Integrals Cheat Sheet
In one line: multiple integrals extend integration to functions of two or more variables, letting us compute volumes, masses, and averages over regions rather than intervals.
Key Ideas
1Double Integrals. Integral over a region R of f(x,y) dA computes the volume under a surface f(x,y) above a 2D region R.
2Iterated Integrals. A double integral is usually computed as two single integrals done one after another — integrate with respect to one variable first, treating the other as constant, then integrate the result.
3Order of Integration. dx dy or dy dx — the order depends on how the region's boundaries are described; sometimes one order is much easier than the other.
4Setting Up Bounds. Inner bounds may depend on the outer variable (e.g., y from 0 to x), while outer bounds are always constants.
5Applications. Double integrals compute volumes under surfaces, areas of 2D regions, and total mass when density varies across a region.
Worked Examples
Evaluate the double integral of x*y over x from 0 to 2 and y from 0 to 3 (dy dx).
9
Evaluate the double integral of 1 dA over x from 0 to 1 and y from 0 to 2 (this gives the area of the rectangle).
2 (matches the rectangle's area, 1 x 2)
Evaluate the double integral of (x+y) over x from 0 to 1 and y from 0 to 1.
1
Formulas
Integral over R of f(x,y) dA = Integral (outer bounds) of [Integral (inner bounds) of f(x,y) d(inner var)] d(outer var)
Practice Yourself
Evaluate the double integral of 2 dA over x from 0 to 2, y from 0 to 3.
12
Evaluate the double integral of x dA over x from 0 to 2, y from 0 to 1.
2
Evaluate the double integral of y dA over x from 0 to 1, y from 0 to 2.
2