147. Find the area of the region between the curve y = 2x / (1 + x²) and the interval −2 ≤ x ≤ 2 of the x-axis.

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Identify the function and the interval: The function given is \(y = \frac{2x}{1 + x^{2}}\) and the interval is \(-2 \leq x \leq 2\).

Determine the points where the curve intersects the x-axis within the interval by setting \(y = 0\): Solve \(\frac{2x}{1 + x^{2}} = 0\) to find the roots.

Split the integral at the points where the function crosses the x-axis to handle areas above and below the x-axis separately, ensuring the total area is positive.

Set up the definite integral(s) for the area between the curve and the x-axis: Calculate \(\int |y| \, dx\) over the appropriate subintervals, which means integrating \(\left| \frac{2x}{1 + x^{2}} \right|\) from \(-2\) to \(2\).

Evaluate the integral(s) by integrating \(\frac{2x}{1 + x^{2}}\) or its negative depending on the sign of \(y\) in each subinterval, then sum the results to find the total 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, especially when the function crosses the x-axis, one must consider the absolute value of the function or split the integral at points where the function changes sign.

Finding Points of Intersection with the x-axis

To determine where the curve crosses the x-axis, set y = 0 and solve for x. These points divide the interval into subintervals where the function is either positive or negative, which is essential for correctly calculating the total area.

Integration of Rational Functions

The given function is a rational function of the form 2x/(1 + x²). Integrating such functions often involves substitution methods, such as letting u = 1 + x², to simplify the integral and find an antiderivative.