Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.
y = x, y = 1/x², x = 2

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First, identify the region enclosed by the curves \(y = x\), \(y = \frac{1}{x^2}\), and the vertical line \(x = 2\). To do this, find the points of intersection between \(y = x\) and \(y = \frac{1}{x^2}\) by solving the equation \(x = \frac{1}{x^2}\).

Solve \(x = \frac{1}{x^2}\) by multiplying both sides by \(x^2\) to get \(x^3 = 1\). Then find the real root(s) of this equation to determine the \(x\)-coordinate(s) of the intersection point(s).

Determine which curve is on top and which is on the bottom between the points of intersection and up to \(x = 2\). This will help set up the integral for the area between the curves correctly.

Set up the definite integral for the area of the enclosed region as \(\int_{a}^{2} \left( \text{top function} - \text{bottom function} \right) \, dx\), where \(a\) is the \(x\)-value of the intersection point found earlier.

Evaluate the integral by integrating the difference of the functions \(y = x\) and \(y = \frac{1}{x^2}\) over the interval \([a, 2]\) to find the area of the enclosed region.

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주요 개념

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

Finding Points of Intersection

To determine the enclosed region, first find where the curves intersect by solving equations simultaneously. For y = x and y = 1/x², set x = 1/x² and solve for x. These intersection points define the limits of integration for calculating the area.

Setting Up Definite Integrals for Area

The area between curves is found by integrating the difference of the functions over the interval between intersection points. Identify which function is on top and subtract the lower function, then integrate with respect to x from the left to right boundary.

Handling Vertical Boundaries in Integration

When a vertical line like x = 2 bounds the region, it sets a fixed limit for integration. Ensure the integral limits reflect this boundary, and if the region is split by intersection points, consider breaking the integral into parts accordingly.

Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.y = x, y = 1/x², x = 2