Problem 37 - Essential Calculus

Question

A region bounded by the parabola \(y=4 x-x^{2}\) and the \(x\) -axis is revolved about the \(x\) -axis. A second region bounded by the parabola \(y=4-x^{2}\) and the \(x\) -axis is revolved about the \(x\) -axis. Without integrating, how do the volumes of the two solids compare? Explain.

Step-by-Step Solution

Answer

The volumes of the two solids formed by rotating the given parabolas about the x-axis are the same.

Step 1: Identify the Parabolas

Look at both parabolas. The first one is \(y=4 x-x^{2}\), which is a parabola facing downwards and symmetrical about the y-axis, and the second one is \(y=4-x^{2}\), a downwards facing parabola which is also symmetrical about the y-axis.

Step 2: Analyze the Cross-Sections

Since these parabolas are revolved around the x-axis, their cross sections perpendicular to the x-axis are circular. Even though the heights of the parabolas are different, the cross-sectional areas at corresponding x-values are identical, given the symmetry.

Step 3: Apply Cavalieri's Theorem

According to Cavalieri's theorem, two solids with identical cross-sectional areas along a height have the same volume. Since the cross-sections of our two volume-forming parabolas at the same x-values are the same (as discussed in step 2), their volumes are also the same.