Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.
y² = 4x, y = 4x - 2

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First, identify the curves given: the parabola \(y^{2} = 4x\) and the line \(y = 4x - 2\).

Find the points of intersection between the two curves by substituting \(y = 4x - 2\) into \(y^{2} = 4x\). This gives the equation \((4x - 2)^{2} = 4x\).

Expand and simplify the equation \((4x - 2)^{2} = 4x\) to find the values of \(x\) where the curves intersect. Solve the resulting quadratic equation for \(x\).

Once the \(x\)-values of the intersection points are found, calculate the corresponding \(y\)-values using \(y = 4x - 2\) to get the exact points of intersection.

Set up the integral for the area between the curves by integrating with respect to \(x\) from the smaller to the larger intersection \(x\)-value. The integrand is the difference between the upper curve and the lower curve, which means integrating \([(4x - 2) - y_{\text{lower}}] \, dx\), where \(y_{\text{lower}}\) is expressed in terms of \(x\) from the parabola.

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Finding Points of Intersection

To determine the enclosed region, first find where the curves intersect by solving their equations simultaneously. This involves substituting one equation into the other and solving for the variable(s), which gives the limits of integration for the area calculation.

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Critical Points

Setting Up the Integral for Area

The area between two curves is found by integrating the difference of their functions over the interval defined by their points of intersection. Depending on the orientation, you may integrate with respect to x or y, ensuring the upper function minus the lower function is used.

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Finding Area When Bounds Are Not Given

Converting and Handling Curves

When dealing with curves like y² = 4x, it may be necessary to express y explicitly as functions of x or vice versa. Understanding how to manipulate and interpret such curves is essential to correctly set up the integral and determine which function bounds the region above or below.

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