Double and Triple Integrals

≈ 25 min

A double integral over a region sums a function across a two-dimensional area:

\iint_R f(x, y)\, dA.

Over a rectangle you compute it as an iterated integral — integrate the inner variable first (treating the other as constant), then the outer.

Worked example. $\displaystyle\int_0^1 \int_0^2 (x + y)\, dy\, dx.$

Inner (in $y$ , with $x$ fixed): $\displaystyle\int_0^2 (x + y)\, dy = \left[ xy + \tfrac{y^2}{2} \right]_0^2 = 2x + 2.$

Outer (in $x$ ): $\displaystyle\int_0^1 (2x + 2)\, dx = \left[ x^2 + 2x \right]_0^1 = 3.$

A triple integral $\iiint_V f\, dV$ extends the same idea to a solid — for example, the mass of a body whose density varies from point to point.

Fubini's theorem says that for a continuous function over a rectangle, swapping the order of integration…

gives the same value

negates the value

always gives zero

only works if $f \geq 0$

Evaluate $\displaystyle\int_0^1 \int_0^1 xy \, dy \, dx.$

Evaluate $\displaystyle\int_0^1 \int_0^2 (x + y) \, dy \, dx.$

A unit cube occupies $0 \le x, y, z \le 1$ with density $\rho(x, y, z) = z$ . Find its mass $\displaystyle\iiint_V z \, dV.$