AP Calculus AB · Applications of integration
Let R be the region bounded by y=4-x^2 and y=x+2
Problem
Let \(R\) be the region bounded by \(y=4-x^{2}\) and \(y=x+2\). (a) Find the \(x\)-coordinates of the points of intersection, and determine which graph is above the other between those \(x\)-values. (b) Find the exact area of \(R\). (c) The region \(R\) is the base of a solid whose cross-sections perpendicular to the \(x\)-axis are squares with sides in \(R\). Find the exact volume of the solid. (d) The region \(R\) is revolved about the horizontal line \(y=-1\). Using washers with respect to \(x\), find the exact volume of the resulting solid. Identify the outer and inner radii as functions of \(x\).
Hint
Subtract the line from the parabola for area and for square sides; shift both graphs toward the offset axis for radii.
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