Basics of Double Integrals

In a Nut Shell: The single integral of a function, y(x), ∫ y dx gives the area under the

function y(x). Likewise the integral of a function, x(y), ∫ x dy also gives the area under

the function, x(y). In these cases the element of area, dA = y dx or dA = x dy.

In a Nut Shell: For double integrals, the element of area, dA, can also be represented by

dA = dx dy or by dA = dy dx. This representation gives rise to a “double integral”.

The value of the integral then becomes A = ∫ ∫ dx dy or A = ∫ ∫ dy dx depending on the

“order” of integration. Depending on the function or functions one order (i.e. integrate x

first followed by y or vice-versa) of integration may be easier than the other order. The

order of integration is optional.

The most convenient order of integration depends on the type of region involved. A Type 1

region occurs when integration with respect to the y-coordinate comes first followed by

integration in the x-direction. The figures below show Type 1 regions. You can think of

the process as “sweeping out” the entire domain using the element of area, dA. Here you

“sweep” the entire domain, D, first in the y-direction followed by the x-direction.

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