Area: Find the area of the region bounded above by y = 2 cos x and below by y = sec x, −π/4 ≤ x ≤ π/4.

검증된 단계별 안내

First, identify the curves that bound the region: the upper curve is given by \(y = 2 \cos x\) and the lower curve by \(y = \sec x\) over the interval \(-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}\).

Next, set up the integral for the area between the two curves. The area \(A\) is given by the integral of the difference between the upper and lower functions over the given interval: \(A = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \left( 2 \cos x - \sec x \right) \, dx\).

Before integrating, verify that \(2 \cos x\) is indeed greater than \(\sec x\) on the interval \([-\frac{\pi}{4}, \frac{\pi}{4}]\) to ensure the correct order of subtraction for the area calculation.

Proceed to integrate each term separately: \(\int 2 \cos x \, dx\) and \(\int \sec x \, dx\). Recall that the integral of \(\cos x\) is \(\sin x\), and the integral of \(\sec x\) is \(\ln |\sec x + \tan x|\).

Finally, evaluate the definite integral by substituting the limits \(x = \frac{\pi}{4}\) and \(x = -\frac{\pi}{4}\) into the antiderivative expression, and subtract the results to find the area.

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주요 개념

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

Definite Integrals for Area Calculation

Definite integrals compute the net area between a curve and the x-axis over a specific interval. When finding the area between two curves, the integral of the difference of the functions over the interval gives the bounded area.

Understanding and Comparing Trigonometric Functions

The problem involves trigonometric functions y = 2 cos x and y = sec x. Knowing their behavior, values, and relative positions on the interval helps determine which function is above or below, essential for setting up the integral correctly.

Interval and Domain Considerations

The interval −π/4 ≤ x ≤ π/4 restricts the domain for integration. It's important to verify that both functions are defined and continuous on this interval to ensure the integral is valid and the area calculation is accurate.