Double Integrals http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQu...
In calculus of a single variable the deﬁnite integral
for f(x)>=0 is the area under the curve f(x) from x=a to x=b. For general f(x) the deﬁnite integral is equal to
the area above the x-axis minus the area below the x-axis.
The deﬁnite integral can be extended to functions of more than one variable. Consider a function of 2
variables z=f(x,y). The deﬁnite integral is denoted by
where R is the region of integration in the xy-plane. For positive f(x,y), the deﬁnite integral is equal to the
volume under the surface z=f(x,y) and above xy-plane for x and y in the region R. This is shown in the ﬁgure
For general f(x,y), the deﬁnite integral is equal to the volume above the xy-plane minus the volume below the
xy-plane. This page includes the following sections:
Brief Discussion of Riemann Sums
Double Integrals over Rectangular Regions
Double Integrals over General Regions
Double integrals arise in a number of areas of science and engineering, including computations of Double Integrals http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQu...
Area of a 2D region
Mass of 2D plates
Force on a 2D plate
Average of a function
Center of Mass and Moment of Inertia
Brief Discussion of Riemann Sums
As in the case of an integral of a function of one variable, a double integral is deﬁned as a limit of a Riemann
sum. Suppose we subdivide the region R into subrectangles as in the ﬁgure below (say there are M rectangles
in the x direction and N rectangles in the y direction).
Label the rectangles R_ij where 1<=i<=M and 1<=j<=N. Think of the deﬁnite integral as representing
volume. The volume under the surface above rectangle R_ij is approximately f(x_i,y_j)A_ij, where A_ij is
area of the rectangle and f(x_i,y_j) is the approximate height of the surface in the rectangle. Here (x_i,y_j) is
some point in the rectangle R_ij. If we sum over all rectangles we have
In the limit as the size of the rectangles goes to 0, the sum on the right converges to a value which is the
deﬁnite integral. The quantity f(x,y)dAin the deﬁnite integral represents the volume in some inﬁnitesimal
region around the point (x,y). The region is so small that the f(x,y) only varies inﬁnitesimally in the region.
The double integral sign says: add up volumes in all the small regions in R.
Double Integrals over a Rectangular Region
Suppose that f(x,y) is continuous on a rectangular region in the xy plane as shown above. The double integral Double Integrals http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQu...
represents the volume under the surface. We can compute the volume by slicing the three-dimensional region
like a loaf of bread. Suppose the slices are parallel to the y-axis. An example of slice between x and x+dx is
shown in the ﬁgure.
In the limit of inﬁnitesimal thickness dx, the volume of the slice is the product of the cross-sectional area and
the thickness dx. The cross sectional area is the area under the curve f(x,y) for ﬁxed x and y varying between
c and d. (Note that if the thickness dx is inﬁnitesimal, x varies only inﬁnitesimally on the slice. We can
assume that x is constant.) The picture below shows the cross-sectional area.
The area is given by the integral Double Integrals http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQu...
The variable of integation is y and x is a CONSTANT. The cross-sectional area depends on x and this is why
we write C=C(x). The volume of the slice between x and x+dx is C(x)dx. The total volume is the sum of the
volumes of all the slices between x=a and x=b:
If substitute for C(x), we obtain:
This is an example of an iterated integral. One integrates with respect to y ﬁrst, then x. The