Average Value: Find the average value of the function f(x) = 1 / (1 - sin θ) on the interval [0, π/6].

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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 given: \( f(\theta) = \frac{1}{1 - \sin \theta} \), with \( a = 0 \) and \( b = \frac{\pi}{6} \).

Set up the integral for the average value: \[ \frac{1}{\frac{\pi}{6} - 0} \int_0^{\frac{\pi}{6}} \frac{1}{1 - \sin \theta} \, d\theta = \frac{6}{\pi} \int_0^{\frac{\pi}{6}} \frac{1}{1 - \sin \theta} \, d\theta \]

To evaluate the integral \( \int \frac{1}{1 - \sin \theta} \, d\theta \), use a trigonometric identity or substitution. One common approach is to multiply numerator and denominator by the conjugate \( 1 + \sin \theta \) to simplify the integrand.

After simplifying, integrate the resulting expression with respect to \( \theta \) from 0 to \( \frac{\pi}{6} \), then multiply by \( \frac{6}{\pi} \) to find 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 accumulated area under the curve of a function between two points a and b. It is essential for finding the total accumulation or average value over an interval.

Integration of Trigonometric Functions

Integrating functions involving trigonometric expressions like 1/(1 - sin θ) often requires substitution or trigonometric identities. Understanding these techniques helps simplify and evaluate the integral accurately.