## The Calculus Primer (2011)

### Part XVI. Successive and Partial Integration; Approximate Integration

### Chapter 60. MULTIPLE INTEGRALS

**16—1. Successive Integration.** This is the inverse of the process of successive differentiation. Suppose it is given that

and we wish to find *y.* We may then write:

integrating:

Again:

integrating once more:

Finally:

and integrating,

The above analysis can also be written as follows:

These last two are called a *double integral* and a *triple integral,* respectively. It will be seen that there is nothing new about successive integration, except that more than one constant of integration is involved. In general, a *multiple*integral requires two or more successive integrations. The process is also known as repeated integration, or *iterated* integration.

EXAMPLE. Find *y*, if *y* = 3*x*^{2} *dx dx dx*.

*Solution.*

*y = * 3*x*^{2} *dx dx dx*

*= * (*x*^{3} + *C*_{1}) *dx dx*

**16—2. Multiple Integrals with Limits of Integration.** If successive integrations are performed between limits, the constants of integration disappear.

EXAMPLE. Evaluate 3*x dx dx dx*.

*Solution.* Beginning by integrating the “inside” integral first:

**EXERCISE 16—1**

**16—3. Successive Partial Integration.** Just as we can find partial derivatives of a function of two or more variables, so we can also integrate the function *f*(*x,y*) in an analogous inverse process of partial differentiation. In the function *f*(*x*,*y*)*,* where *x* and *y* are both independent variables, let us for a moment consider *x* as a constant, and let *y* vary; then *f*(*x,y*) becomes a function of *y* only. Now, under these conditions, suppose we integrate between the limits *y* = *c* and *y* = *d*; we then have:

Now the value of this integral will depend not only upon the value of *y,* but also upon the value of *x*; hence the entire expression in (1) may be regarded as a function of *x.* Under this condition, let us now integrate with respect to *x*between the limits *x* = *a* and *x* = *b*: the result becomes

which is generally written without the bracket as

The expression (3) is read: “*the double integral of f*(*x,y*) *from y* = *c to y* = *d and from x* = *a to x* = *b*.”

EXAMPLE 1. Find the value of the double integral

*Solution.* We perform the “*y*-integration” first, remembering to “hold” *x* constant:

Now we perform the second integration, or the “*x*-integration,” upon the expression in (1), this time “holding” *y* constant, and integrating with respect to *x*:

The limits of integration need not necessarily all be constants; very often the limits of *y* in the first integration are themselves functions of the variable *x,* as shown in the next two examples.

EXAMPLE 2. Find the value of

*Solution*. Integrating first with respect to *y*, we get:

Now, integrating with respect to *x*:

EXAMPLE 3. Find

*Solution*.

The same ideas may be extended to *triple integrals*.

EXAMPLE 4. Find the value of

*Solution*. Consider *x* and *y* both constant, and integrate with respect to the variable *z*:

Then perform the *y*-integration, remembering that *x* is a constant:

Finally, perform the *x*-integration:

**EXERCISE 16—2**

*Find the value of each of the following multiple integrals*: