Determine the area of the shaded region in the following figures.

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Identify the curves that bound the shaded region. The region is bounded by the line \(x = 2y\) and the curve \(x = y^2 - 3\).

Determine the points of intersection by setting the two expressions for \(x\) equal: \(2y = y^2 - 3\). Rearrange this to form a quadratic equation: \(y^2 - 2y - 3 = 0\).

Solve the quadratic equation \(y^2 - 2y - 3 = 0\) to find the \(y\)-values of the intersection points. These will be the limits of integration.

Set up the integral for the area of the shaded region with respect to \(y\). The area is given by the integral of the difference between the rightmost curve and the leftmost curve: \(\int_{y_1}^{y_2} [(2y) - (y^2 - 3)] \, dy\).

Evaluate the integral to find the area. This involves integrating the function \(2y - y^2 + 3\) over the interval from \(y_1\) to \(y_2\).

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주요 개념

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

Finding Points of Intersection

To determine the area between two curves, first find their points of intersection by solving their equations simultaneously. These points define the limits of integration and ensure the correct region is considered.

Setting Up the Integral for Area Between Curves

The area between two curves is found by integrating the difference of the functions over the interval defined by their intersection points. When curves are given as x in terms of y, integrate with respect to y, subtracting the left curve from the right.

Integrating with Respect to y

Since the curves are expressed as x = f(y), the integral for the area is set up with respect to y. This involves integrating the horizontal distance between the curves (right minus left) over the y-interval, which may differ from the usual vertical integration.