Multiple Integrals
Multiple Integrals are basically a continuation of regular integrals that we already went over in AP Calc. Throughout the year, we reviewed only integrals that have a single variable. Multiple integrals have more than one variable, usually 2 or 3.
Instead of calculating the area under the curve of a function on a 2-dimentional plane, an integral with 2 variables calculates the volume that a certain function makes on a 3 dimensional plane, as shown below:
Since an integral with 2 variables can be used to calculate the 3rd dimension, Triple integrals can be used to calculate a 4th dimension, such as time or density.
We can talk all we want about what integrals are used for, but it is easier to understand just by doing a couple examples:
We will start with a simple example,
When
We can check our work by flipping the limits on the integral, and then also flipping the order of integration. But you always have to Integrate inside out.
Bingo! The answers are the same.
Next, we will try integrating this same function, except our limits will be variables instead of constants. This seems complicated, but is actually quite easy.
To tackle multiple integrals, remember to take it one step at a time until you feel comfortable speeding up.
To watch another example by Patrick JMT, watch
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