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Calculus (Guichard)
9: Applications of Integration

9: Applications of Integration

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Feb 8, 2025
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- 8.E: Techniques of Integration (Exercises)
- 9.1: Area Between Curves

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9.1: Area Between Curves
9.2: Distance, Velocity, and Acceleration
9.3: Volume
We have seen how to compute certain areas by using integration; some volumes may also be computed by evaluating an integral. Generally, the volumes that we can compute this way have cross-sections that are easy to describe.
9.4: Average Value of a Function
The average of some finite set of values is a familiar concept. It can be directly connected to an application of an integral.
9.5: Work
A fundamental concept in classical physics is work: If an object is moved in a straight line against a force $F$ for a distance $s$ the work done is $W=Fs$ .
9.6: Center of Mass
The center of mass of an object is the point that represents the mean position of the matter in a body or system.
9.7: Kinetic energy and Improper Integrals
an integral, with a limit of infinity, is called an improper integral. If the value of an improper integral is a finite number, as in this example, we say that the integral converges, and if not we say that the integral diverges.
9.8: 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.
9.9: 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.
9.10: 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.
9.E: Applications of Integration (Exercises)
These are homework exercises to accompany David Guichard's "General Calculus" Textmap.

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David Guichard (Whitman College)

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