Finding area
Find the area of the region enclosed by the curve y = x cos(x) and the x-axis (see the accompanying figure) for:
b. 3π/2 ≤ x ≤ 5π/2.

검증된 단계별 안내

Identify the region bounded by the curve $y = x \cos(x)$ and the x-axis for the interval $\frac{3\pi}{2} \leq x \leq \frac{5\pi}{2}$. Notice from the graph that the curve crosses the x-axis at $x = \frac{3\pi}{2}$ and $x = \frac{5\pi}{2}$, and the function is negative between these points.

Since the function is below the x-axis in this interval, the area between the curve and the x-axis is given by the integral of the absolute value of the function. This means we need to integrate $-y = -x \cos(x)$ over $\left[ \frac{3\pi}{2}, \frac{5\pi}{2} \right]$ to get a positive area.

Set up the integral for the area as $\displaystyle A = \int_{\frac{3\pi}{2}}^{\frac{5\pi}{2}} -x \cos(x) \, dx$.

Use integration by parts to evaluate the integral. Let $u = x$ and $dv = \cos(x) dx$. Then, compute $du = dx$ and $v = \sin(x)$. Apply the integration by parts formula: $\int u \, dv = uv - \int v \, du$.

After finding the antiderivative, evaluate it at the bounds $x = \frac{5\pi}{2}$ and $x = \frac{3\pi}{2}$, then subtract to find the definite integral value, which represents the area of the region.

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주요 개념

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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. When the function crosses the x-axis, the integral sums positive and negative areas, so to find the total enclosed area, you must consider the absolute value or split the integral at zeros.

Finding Zeros of the Function

To determine the area enclosed by the curve and the x-axis, it is essential to find where the function crosses the x-axis (i.e., where y = 0). For y = x cos(x), zeros occur when cos(x) = 0 or x = 0, which helps in splitting the integral into intervals where the function is positive or negative.

Properties of the Function y = x cos(x)

The function y = x cos(x) oscillates with increasing amplitude as x increases, crossing the x-axis multiple times. Understanding its behavior between 3π/2 and 5π/2, including the sign of the function, is crucial for correctly setting up the integral to find the enclosed area.

Finding areaFind the area of the region enclosed by the curve y = x cos(x) and the x-axis (see the accompanying figure) for:b. 3π/2 ≤ x ≤ 5π/2.