What does a double integral represent?

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It can be difficult to visualize what a double integral represents, which is why in this video we’ll be answering the question, “What am I finding when I evaluate a double integral?”

In order to answer this question, we’ll compare the double integral to a single integral, so that we understand exactly how to transition from single variable calculus into multivariable calculus. Every piece of the single integral, like the integral, the bounds or limits of integration, the function which is the integrand, and the differential (usually dx) will all translate into a corresponding piece of the double integral.

If we want to describe these with words, we can say that for the single integral, we’re integrating a single variable function f(x) over the interval [a,b], using vertical slices of area, in order to find the total area under the curve f(x) but above the x-axis.

In contrast, we can say that for the double integral, we’re integrating a multivariable function f(x,y) over the region R which is defined for x on the interval [a,b] and for y on the interval [c,d], using vertical slices of volume, in order to find the total volume under the surface f(x,y) but above the xy-plane.

Skip to section:

For the single integral:
0:36 // Sketching a single variable integral
6:32 // Building the area equation for a single integral
7:10 // Moving from the geometric estimation to summation notation
9:24 // Moving from summation notation to the single integral

For the double integral:
10:39 // Sketching a multivariable double integral
20:38 // Building the volume equation for a double integral
21:16 // Moving from the geometric estimation to summation notation
23:22 // Moving from summation notation to the double integral

24:25 // Summary

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