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Mathematics LibreTexts

3: Multiple Integrals

The multiple integral is a generalization of the definite integral with one variable \[\int f(x) dx\] to functions of more than one real variable, for example, \[\iint f(x, y) \,dx\,dy\] for an indefinite double integral or \[\iiint f(x, y, z)\, dx\,dy\,dz\] for an indefinite  triplet integral. For definite multiple integrals, each variable can have different limits of integration.

  • 3.1: Double Integrals
    In single-variable calculus, differentiation and integration are thought of as inverse operations. There is a similar way of defining integration of real-valued functions of two or more variables? Recall also that the definite integral of a nonnegative function f(x)≥0 represented the area “under” the curve y=f(x). As we will now see, the double integral of a nonnegative real-valued function f(x,y)≥0 represents the volume “under” the surface z=f(x,y).
  • 3.2: Double Integrals Over a General Region
    Previously, we got an idea of what a double integral over a rectangle represents. We can now define the double integral of a real-valued function f(x,y) over more general regions in R2.
  • 3.3: Triple Integrals
    While the double integral could be thought of as the volume under a two-dimensional surface. It turns out that the triple integral simply generalizes this idea: it can be thought of as representing the hypervolume under a three-dimensional hypersurface in R4 . In general, the word “volume” is often used as a general term to signify the same concept for anynn -dimensional object (e.g. length in R1 , area in R2 ).
  • 3.4: Numerical Approximation of Multiple Integrals
    For complicated functions, it may not be possible to evaluate one of the iterated integrals in a simple closed form. Luckily there are numerical methods for approximating the value of a multiple integral. The method we will discuss is called the Monte Carlo method. The idea behind it is based on the concept of the average value of a function, which you learned in single-variable calculus.
  • 3.5: Change of Variables in Multiple Integrals
    Given the difficulty of evaluating multiple integrals, the reader may be wondering if it is possible to simplify those integrals using a suitable substitution for the variables. The answer is yes, though it is a bit more complicated than the substitution method which you learned in single-variable calculus.
  • 3.6: Application: Center of Mass
    The center of mass of a distribution of mass in space is the unique point where the weighted relative position of the distributed mass sums to zero or the point where if a force is applied causes it to move in direction of force without rotation. The distribution of mass is balanced around the center of mass and the average of the weighted position coordinates of the distributed mass defines its coordinates. Calculations in mechanics are often simplified by using center of mass formulations.
  • 3.7: Application: Probability and Expectation Values
    In this section we will briefly discuss some applications of multiple integrals in the field of probability theory. In particular we will see ways in which multiple integrals can be used to calculate probabilities and expected values.
  • 3.E: Multiple Integrals (Exercises)
    Problems and select solutions to the chapter.

Thumbnail: Double integral as volume under a surface \(z = 10 − x^2 − y^2/8\). The rectangular region at the bottom of the body is the domain of integration, while the surface is the graph of the two-variable function to be integrated. Image used with permission (Public Domain; Oleg Alexandrov).