66-68. Areas of regions (Use of Tech) Find the area of the following regions.
66. The region bounded by the curve y = (x - x²)/[(x + 1)(x² + 1)] and the x-axis from x = 0 to x = 1

검증된 단계별 안내

Identify the function to integrate: \( y = \frac{(x - x^{2})}{(x + 1)(x^{2} + 1)} \). We want to find the area between this curve and the x-axis from \( x = 0 \) to \( x = 1 \).

Set up the definite integral for the area: \[ \text{Area} = \int_{0}^{1} \left| \frac{(x - x^{2})}{(x + 1)(x^{2} + 1)} \right| \, dx \]. Since area is always positive, we take the absolute value of the function inside the integral.

Determine where the function \( y \) changes sign on the interval \( [0,1] \) by solving \( y = 0 \). This means solving \( x - x^{2} = 0 \) because the denominator is always positive for \( x \geq 0 \). The roots are \( x = 0 \) and \( x = 1 \), so check the sign of \( y \) between these points to see if the function is positive or negative.

If the function does not change sign between 0 and 1, you can remove the absolute value and integrate \( y \) directly. Otherwise, split the integral at the points where \( y = 0 \) and integrate the absolute value accordingly.

Use technology (such as a graphing calculator or computer algebra system) to evaluate the definite integral(s) set up in the previous step to find the exact area.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

11m

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 [a, b] represents the net area between the curve and the x-axis. When the function is positive, the integral gives the area directly; if negative, it subtracts area. To find the total area bounded by the curve and the x-axis, one may need to consider absolute values or split the integral at points where the function crosses the axis.

Analyzing the Sign of the Function

Before integrating, it is important to determine where the function is positive or negative within the interval. This helps in correctly calculating the area, as areas below the x-axis contribute negatively to the integral. Identifying zeros of the function within the interval allows splitting the integral accordingly.

Use of Technology in Integration

Complex rational functions like y = (x - x²)/[(x + 1)(x² + 1)] may be difficult to integrate by hand. Technology tools such as graphing calculators or computer algebra systems can assist in evaluating definite integrals accurately and efficiently, as well as in visualizing the function to understand its behavior over the interval.

66-68. Areas of regions (Use of Tech) Find the area of the following regions.66. The region bounded by the curve y = (x - x²)/[(x + 1)(x² + 1)] and the x-axis from x = 0 to x = 1