Double Integration: Method, Formulas & Examples

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  • 0:05 What are Double Integrals?
  • 2:14 Calculating Double Integrals
  • 2:47 Example Calculation 1
  • 4:49 Example Calculation 2
  • 6:47 Calculation Example 3:…
  • 8:36 Lesson Summary
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Lesson Transcript
Instructor: Shaun Ault

Shaun is currently an Assistant Professor of Mathematics at Valdosta State University as well as an independent private tutor.

In this lesson, we explore the method of double integration, which is useful in finding certain areas, volumes, and masses of objects. Note: knowledge of (single) integrals is a must before tackling double integrals!

What are Double Integrals?

The theory behind integration is long and complex, but you should be familiar with integration as the method for finding the area under a curve (among other important applications). You may recall how an integral is based on approximating area using very thin rectangles. In fact, the dx part of the integral notation is just the width of an approximating rectangle. But what about integrating in higher dimensions?

Consider the volume under a surface with equation z = f (x, y). We can approximate this volume in much the same way we approximate area, by filling the region with incredibly thin and narrow boxes. The length and width of each box may be called dx and dy, while the height is given by the function value f (x, y). Then, after adding up the volumes of many of these boxes over the region R of the plane that serves as the base of the solid (and allowing dx, dy → 0), we get the exact volume.

The notation for this process of finding volume, which is called double integration, is represented by the following notation (see below).

The expression dA = dy dx is called the area element, and it represents the area of the base of each small box. In the case that the region R is the rectangle [a, b] × [c, d], that is, axb and cyd, then we write the bounds of integration on each integral symbol, and expand the area element as indicated.

Equation for Double Integrals
double integral

Calculating Double Integrals

If you can do a single integral, then you can compute a double integral. This method is called iterated integration. Simply tackle each integral from inside to outside. Remember, to evaluate an integral, you have to find an anti-derivative and then plug in the bounds of integration and subtract. The only added wrinkle here is that the first integral is done with respect to the variable y, while letting x be considered a constant. Let's work out an example in order to illustrate the method.

Example Calculation 1

Evaluate the double integral of f (x, y) = 9x^3 y^2 over the region R = [1, 3] × [2, 4].

Example 1

Now what about non-rectangular regions? Well, as long as R can be expressed as the region between two curves, then iterated integration will work beautifully. Let's see an example in action!

Example Calculation 2

Evaluate the double integral of f (x, y) = x + 3y over the region R bounded by y = x^2 and y = x.

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