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8: Applications of Integration

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    186217
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    • 8.1: Arc Length
      Here is another geometric application of the integral: find the length of a portion of a curve. As usual, we need to think about how we might approximate the length, and turn the approximation into an integral.
    • 8.2: Surface Area
      Another geometric question that arises naturally is: "What is the surface area of a volume?'' For example, what is the surface area of a sphere? More advanced techniques are required to approach this question in general, but we can compute the areas of some volumes generated by revolution.
    • 8.3: Moments and Centers of Mass
      In this section, we consider centers of mass (also called centroids, under certain conditions) and moments. The basic idea of the center of mass is the notion of a balancing point. Many of us have seen performers who spin plates on the ends of sticks. The performers try to keep several of them spinning without allowing any of them to drop. Mathematically, that sweet spot is called the center of mass of the plate.
    • 8.4: Probability
      A variable, say \(X\), that can take certain values, each with a corresponding probability, is called a random variable; in the example above, the random variable was the sum of the two dice. If the possible values for X are \(x_1\), \( x_2\(,…,\) x_n\), then the expected value of the random variable is \( E(X)=\sum_{i=1}^n x_iP(x_i)\). The expected value is also called the mean.


    This page titled 8: Applications of Integration is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by David Guichard via source content that was edited to the style and standards of the LibreTexts platform.

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