Find The Average Value Of The Function Over The Given Interv

Find the average value of the function over the given interval

Determine the average value of each specified function over its given interval by applying the average value formula for functions:

The average value \(f_{avg}\) of a function \(f(x)\) on \([a, b]\) is computed as:

\(f_{avg} = \frac{1}{b - a} \int_{a}^{b} f(x) \, dx\)

Calculate this integral accurately, then interpret the result as the average value over the specified interval.

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The calculation of average values of functions over defined intervals is a fundamental concept in integral calculus with applications spanning physics, economics, and engineering. It provides a mean or typical value of the function over a specified domain, offering insights into the overall behavior of the system modeled by the function.

For illustrative purposes, consider the function \(f(x) = 6\sqrt{x}\) over the interval \([0, 4]\). The process begins with applying the formula for average value:

\(f_{avg} = \frac{1}{4 - 0} \int_{0}^{4} 6 \sqrt{x} \, dx = \frac{1}{4} \times 6 \int_{0}^{4} x^{1/2} \, dx\)

The integral of \(x^{1/2}\) is \(\frac{2}{3} x^{3/2}\), so plugging in the limits:

\(6 \times \frac{2}{3} \left[ x^{3/2} \right]_0^4 = 6 \times \frac{2}{3} (4^{3/2} - 0)\)

Evaluating \(4^{3/2}\) simplifies to \(8\), yielding:

\(6 \times \frac{2}{3} \times 8 = 6 \times \frac{16}{3} = \frac{96}{3} = 32\)

Dividing by 4 gives the average value:

\(f_{avg} = \frac{32}{4} = 8\)

Thus, the average value of \(f(x) = 6\sqrt{x}\) over \([0, 4]\) is 8.

Similarly, for the function \(f(x) = e^{x/7}\) over \([0, 7]\):

\(f_{avg} = \frac{1}{7} \int_{0}^{7} e^{x/7} dx\)

Letting \(u = x/7\), so \(dx = 7 du\), the integral becomes:

\(\int_{0}^{1} e^{u} \times 7 du = 7 \int_{0}^{1} e^{u} du = 7 [ e^{u} ]_0^1 = 7 (e^{1} - 1)\)

Calculating:

\(7 (e - 1)\), and dividing by 7 yields: \(e - 1\), so the average value is:

\(f_{avg} = e - 1 \approx 2.718 - 1 \approx 1.718\)

By extending this approach to other functions, one can systematically compute average values over specified intervals.

Calculations of this kind are crucial for understanding the overall behavior of systems modeled by continuous functions, whether it be for predicting average temperature, pollution levels, or economic indicators over time or space.

References

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