Double and Triple Integrals: every key term you need (+ practice quiz)
25 flashcard terms for Multivariable Calculus Topic 6, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 15-question quiz — free, no account needed.
The limit of Riemann sums of function values times small rectangular areas, representing signed volume between the graph and the base rectangle.
Riemann sum in two variables
A finite sum of sample function values weighted by subrectangle areas. Refining the partition in both directions drives the sum toward the double integral.
Integrability of a bounded function
A bounded function on a closed bounded region is integrable if its set of discontinuities has zero area, which covers piecewise continuous cases.
Iterated integral
A nested pair of single integrals evaluated from the inside out, with the inner variable treated as the only variable while outer ones are held constant.
Fubini theorem
For a continuous function on a rectangle, the double integral equals either iterated integral, so the order of integration may be exchanged freely.
Type I region
A plane region bounded above and below by graphs of functions of x over a fixed interval. It is integrated with the y limits inside and constant x limits outside.
Type II region
A plane region bounded left and right by graphs of functions of y over a fixed interval, integrated with the x limits inside and constant y limits outside.
Variable limits rule
Inner limits may depend on the outer variables, but outer limits must be constants. Violating this leaves a variable in the final answer, a reliable sign of error.
Reversing the order of integration
Redescribing the same region with the other variable outermost, often the only way to evaluate an integral whose inner antiderivative has no elementary form.
Sketching the region before integrating
The practical first step in any multiple integral, since limits are read off the picture and reversal is impossible without knowing the region's shape.
Area as a double integral
Integrating the constant one over a plane region returns its area, which gives a useful check on limits before a harder integrand is inserted.
Volume between two surfaces
The double integral of the upper surface minus the lower surface over their common shadow region, valid where the upper truly stays above the lower.
Additivity over subregions
An integral over a region split into nonoverlapping pieces equals the sum of the integrals over the pieces, which is how nonstandard regions are handled.
Average value of a function on a region
The integral of the function over the region divided by the region's area or volume, the natural generalisation of the one-variable mean value.
Triple integral
The limit of sums of function values times small volumes over a solid, computed as three nested single integrals with at most the outermost pair having constant limits.
Volume as a triple integral
Integrating the constant one over a solid returns its volume, again a good sanity check on the limit setup before a density is introduced.
Projection of a solid
The shadow of a solid onto a coordinate plane, which supplies the outer two limits once the inner variable has been bounded by the entering and exiting surfaces.
Order of integration for a solid
Six orders are possible, and a wise choice can turn an intractable integral into a routine one; each order requires a fresh reading of the solid's geometry.
Mass from a density function
The integral of density over the region, reducing to area or volume when the density is constant.
Moment about an axis
The integral of density weighted by the signed distance to that axis, the ingredient needed to locate a centre of mass.
Centre of mass
The point whose coordinates are the corresponding moments divided by the total mass, the balance point of the distributed material.
Centroid
The centre of mass of a region with constant density, depending only on geometry. It need not lie inside a nonconvex region.
Moment of inertia
The integral of density weighted by the square of the distance to an axis, measuring resistance to rotation about that axis.
Jacobian determinant
The determinant of the matrix of partial derivatives of a change of variables, whose absolute value is the local area or volume scaling factor.
General change of variables formula
Rewrites an integral in new variables by substituting the transformed integrand and multiplying by the absolute Jacobian, with the new limits describing the preimage region.