Integration can be used to find the average value of a function over an interval from "summary" of Skills in Mathematics - Integral Calculus for JEE Main and Advanced by Amit M Agarwal
To find the average value of a function over an interval using integration, we need to consider a function f(x) defined on the closed interval [a, b]. The average value of this function over the interval [a, b] can be calculated by dividing the integral of the function over the interval by the length of the interval (b - a). Mathematically, the average value of the function f(x) over the interval [a, b] is given by: \[f_{avg} = \frac{1}{b - a} \int_{a}^{b} f(x) dx\] This formula essentially represents the total area under the graph of the function f(x) over the interval [a, b] divided by the length of the interval. By finding this average value, we can obtain a single representative value that captures the overall behavior of the function over the interval. The concept of finding the average value of a function using integration is particularly useful in various real-world applications where we need to understand the typical behavior or performance of a function over a given interval. For instance, in physics, the average value of a velocity function over a time interval can provide insights into the average speed of an object during that time period. By utilizing integration to find the average value of a function over an interval, we can simplify complex calculations and arrive at a concise summary value that conveys the essential information about the function's behavior over the specified interval. This approach allows us to analyze functions more effectively and extract meaningful insights from their behavior over a given range.
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