Find the total area of the region enclosed by the curve x = y²/³ and the lines x = y and y = -1.

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First, rewrite the given curve and lines clearly: the curve is \(x = y^{2/3}\), and the lines are \(x = y\) and \(y = -1\).

Identify the points of intersection between the curve \(x = y^{2/3}\) and the line \(x = y\) by setting \(y^{2/3} = y\). Solve this equation to find the \(y\)-values where they intersect.

Determine the region enclosed by these curves and lines. Since \(y = -1\) is a horizontal line, find the corresponding \(x\)-values on the curve and line at \(y = -1\) to understand the vertical bounds of the region.

Set up the integral for the area. Because the region is bounded between two curves expressed as \(x\) in terms of \(y\), it is convenient to integrate with respect to \(y\). The area \(A\) can be expressed as \(A = \int_{y=a}^{y=b} (x_{right} - x_{left}) \, dy\), where \(x_{right}\) and \(x_{left}\) are the rightmost and leftmost \(x\)-values at each \(y\).

Determine the limits of integration \(a\) and \(b\) from the intersection points and the line \(y = -1\). Then, write the integral explicitly using the expressions for \(x\) from the curve and the line, and prepare to evaluate it to find the total enclosed area.

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Understanding the Region Bounded by Curves

To find the area enclosed by curves, first identify the region of interest by determining the points of intersection and the boundaries. This involves analyzing the given curves and lines to understand which parts form the enclosed region.

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Finding Area When Bounds Are Not Given

Setting up the Integral with Respect to y

Since the curves are given as functions of y (x in terms of y), it is often easier to integrate with respect to y. The area can be found by integrating the horizontal distance between the curves over the interval defined by the y-boundaries.

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Integration by Parts for Definite Integrals Example 8

Evaluating Definite Integrals for Area Calculation

Once the integral is set up, compute the definite integral by evaluating the difference between the functions over the specified limits. This yields the total enclosed area between the curves and lines.

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