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AP Calculus AB & BC: Help and Review17 chapters | 160 lessons

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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!

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* â†’ 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, *a* â‰¤ *x* â‰¤ *b* and *c* â‰¤ *y* â‰¤ *d,* then we write the **bounds of integration** on each integral symbol, and expand the area element as indicated.

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.

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

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!

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

First, we'll find the points of intersection for the region *R*. Setting *x*^2 = *x*, we find *x* = 0, 1. Now this tells us that *a* = 0 and *b* = 1 are the *x*-bounds for the double integral. You can think of the bottom function *y* = *x^2* as *c,* and the top function *y* = *x* as *d* in the double-integral formula we just looked at.

As mentioned in the intro, double integrals are defined in just the right way to compute the volume of a solid under a given surface *z* = *f* (*x*, *y*), whose base is a given region *R*.

Find the volume of the solid bounded by *x*cos(*xy*) and the *xy*-plane, whose base is the rectangle [0, Ï€/2] × [0, Ï€/2].

Let's set up the double integral and solve to find the volume.

Thus, the volume of the object is *V* = 1.134 cubic units.

Let's review what we've learned here. The notation for this process of finding volume is called **double integration**. You should remember that if you can do a single integral, then you can compute a double integral, which is called **iterated integration**.

A **double integral** is an integral of a two-variable function *f* (*x*, *y*) over a region *R*. If *R* = [*a*, *b*] × [*c*, *d*], then the double integral can be done by iterated integration (integrate first with respect to *y*, and then integrate with respect to *x*).

The notations for double integrals are shown again below.

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

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AP Calculus AB & BC: Help and Review17 chapters | 160 lessons

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