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BackgroundWhen talking about any mathematical quantity, it's important to keep track of three different questions: What is it?How do you compute it? What is it good for? This is especially true for integration, where the three questions have very different answers:
The application of integration that is easiest to understand is area under a curve, which is why we used area to introduce integrals. Likewise, we will use volume to introduce double integrals. But remember that ordinary integrals are about much more than area, and double integrals are about much more than volume. In the following video, we review the ideas and definitions of onedimensional integrals. Not how to compute an integral, but what it is and what it's good for. On the next page, we'll see how the exact same ideas work in two and three dimensions. We'll tackle 'how do you compute it?' in the next learning module. Some applications other than area and volume


These ideas are illustated with the standard `area under a curve' example. The video on the right shows how this area can be approximated by adding up narrower and narrower rectangles, with $x_i^*$ being the left endpoint. (You may need to hit the 'replay' button a few times, since it's only 6 seconds long.) 