Determine whether the following statements are true and give an explanation or counterexample.

b. The area of the region between y=sin x and y=cos x on the interval [0,π/2] is ∫π/20(cosx−sinx)dx.

검증된 단계별 안내

First, identify the two functions given: \(y = \sin x\) and \(y = \cos x\) on the interval \([0, \frac{\pi}{2}]\).

Determine which function is on top (greater) and which is on the bottom over the interval \([0, \frac{\pi}{2}]\). This is important because the area between two curves is found by integrating the difference between the top function and the bottom function.

Evaluate the values of \(\sin x\) and \(\cos x\) at the endpoints: at \(x=0\), \(\sin 0 = 0\) and \(\cos 0 = 1\), so \(\cos x\) is greater; at \(x=\frac{\pi}{2}\), \(\sin \frac{\pi}{2} = 1\) and \(\cos \frac{\pi}{2} = 0\), so \(\sin x\) is greater. This means the two curves cross somewhere in the interval.

Find the point where \(\sin x = \cos x\) in \([0, \frac{\pi}{2}]\) by solving \(\sin x = \cos x\). This will give the exact point where the top and bottom functions switch.

To find the total area between the curves, split the integral at the crossing point and integrate the absolute difference accordingly: \(\int_0^{c} (\cos x - \sin x) \, dx + \int_c^{\frac{\pi}{2}} (\sin x - \cos x) \, dx\), where \(c\) is the crossing point. Therefore, the statement that the area is \(\int_0^{\frac{\pi}{2}} (\cos x - \sin x) \, dx\) without splitting is not correct.

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주요 개념

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

Definite Integral as Area Between Curves

The definite integral of the difference between two functions over an interval gives the net area between their graphs. If one function is always above the other, the integral of (top function - bottom function) over that interval equals the area between them.

Determining Which Function is on Top

To correctly set up the integral for area, identify which function is greater on the interval. For y = sin x and y = cos x on [0, π/2], compare values to see which curve lies above, ensuring the integrand is the top function minus the bottom function.

Evaluating the Integral and Sign of the Result

The integral ∫ (cos x - sin x) dx over [0, π/2] may yield a positive or negative value depending on which function is on top. A negative result indicates the order of subtraction is reversed, so the absolute value or swapping the order is needed to represent the area.

Determine whether the following statements are true and give an explanation or counterexample.b. The area of the region between y=sin x and y=cos x on the interval [0,π/2] is ∫π/20(cosx−sinx)dx.