Find the average value of f(x) = (√(x + 1)) / √x on the interval [1, 3].

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Recall the formula for the average value of a function \(f(x)\) on the interval \([a, b]\): \[\text{Average value} = \frac{1}{b - a} \int_a^b f(x) \, dx\]

Identify the function and interval: here, \(f(x) = \frac{\sqrt{x + 1}}{\sqrt{x}}\) and the interval is \([1, 3]\). So, \(a = 1\) and \(b = 3\).

Rewrite the function to simplify the integral: \[f(x) = \frac{\sqrt{x + 1}}{\sqrt{x}} = \sqrt{\frac{x + 1}{x}} = \sqrt{1 + \frac{1}{x}}\]

Set up the integral for the average value: \[\frac{1}{3 - 1} \int_1^3 \sqrt{1 + \frac{1}{x}} \, dx = \frac{1}{2} \int_1^3 \sqrt{1 + \frac{1}{x}} \, dx\]

To evaluate the integral, consider substitution or algebraic manipulation to simplify the integrand before integrating. After finding the integral, multiply by \(\frac{1}{2}\) to get the average value.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Average Value of a Function

The average value of a function f(x) on an interval [a, b] is given by (1/(b - a)) times the definite integral of f(x) from a to b. It represents the mean height of the function over that interval.

Definite Integral

A definite integral calculates the net area under the curve of a function between two points a and b. It is essential for finding the total accumulation or average value of a function over an interval.

Simplifying and Integrating Radical Functions

Functions involving square roots often require algebraic manipulation to simplify before integration. Techniques include rewriting radicals as fractional exponents and applying integration rules for power functions.