Finding area
Find the area of the region enclosed by the curve y = x sin(x) and the x-axis (see the accompanying figure) for:
c. 2π ≤ x ≤ 3π.

검증된 단계별 안내

Identify the function and the interval: The function given is $y = x \sin x$, and we need to find the area enclosed by this curve and the x-axis for $2\pi \leq x \leq 3\pi$.

Analyze the behavior of the function on the interval: From the graph, observe that $y = x \sin x$ is positive on the interval $2\pi \leq x \leq 3\pi$ because $\sin x$ is positive in this interval and $x$ is positive as well.

Set up the integral for the area: Since the function is above the x-axis in this interval, the area is given by the definite integral of the function from $2\pi$ to $3\pi$:

\[\text{Area} = \int_{2\pi}^{3\pi} x \sin x \, dx\]

To solve this integral, use integration by parts where you let $u = x$ and $dv = \sin x \, dx$. Then compute $du = dx$ and $v = -\cos x$. Apply the integration by parts formula and evaluate the resulting expression at the bounds $2\pi$ and $3\pi$ to find the area.

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주요 개념

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

Definite Integral for Area Calculation

The definite integral of a function over an interval gives the net area between the curve and the x-axis. To find the total area enclosed, you integrate the absolute value of the function or split the integral at points where the function crosses the x-axis.

Behavior of the Function y = x sin(x)

The function y = x sin(x) oscillates due to the sine component, with amplitude increasing linearly because of the x multiplier. Understanding where the function crosses the x-axis (roots) helps determine intervals where the function is positive or negative, which is crucial for area calculation.

Handling Areas Below the x-axis

When the curve lies below the x-axis, the definite integral yields a negative value. To find the actual area, you take the absolute value of the integral over those intervals or split the integral at zeros and sum the absolute values of each part.

Finding areaFind the area of the region enclosed by the curve y = x sin(x) and the x-axis (see the accompanying figure) for:c. 2π ≤ x ≤ 3π.