AP Calculus BC · Contextual applications of differentiation
Let R be the region bounded by the curves x=y^2-2 and x=4
Problem
Let \(R\) be the region bounded by the curves \(x=y^{2}-2\) and \(x=4\). (a) Find the \(y\)-coordinates of the points of intersection, and determine which curve is to the right of the other between those \(y\)-values. (b) Find the exact area of \(R\) by integrating with respect to \(y\). Identify the right and left boundaries as functions of \(y\). (c) The same region can be described with respect to \(x\) as the set of points between a lower graph \(y=-u(x)\) and an upper graph \(y=u(x)\). Find the formula for \(u(x)\), the corresponding interval of \(x\)-values, and a single definite integral with respect to \(x\) for the area of \(R\). Evaluate the integral and confirm that it agrees with (b).
Hint
A left-opening parabola cut by a vertical line is most naturally integrated in \(y\). The \(x\)-description is the pair of square-root branches.
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