Mathematics

Problem

Consider the region

T

bounded by the curves

y = x

and

y = x^{2}

between

x = 0

and

x = 3

. The region is revolved about the horizontal line

y = 1

, which lies between the two curves for some portion of the interval. (a) Determine all values of

x

[0,3]

at which either curve intersects the line

y = 1

. Explain how these values will affect the setup of any integral used to compute the volume of the solid generated. (b) Write a piecewise–defined definite integral (or sum of definite integrals) that represents the exact volume of the solid obtained by the rotation of

T

about

y = 1

. Clearly indicate the intervals over which each expression applies. (c) Evaluate your integral(s) from part (b) to obtain the exact volume.

Intuitive explanation

The key is recognizing that when you revolve around a horizontal line that's inside the region, you create a washer—not a solid disk. For each $x$ -slice, the outer radius is the distance from $y=1$ to the farther curve, and the inner radius is the distance to the closer curve. You'll need to identify where $y=x$ and $y=x^2$ swap positions relative to the axis of rotation, then set up the washer integral with $\pi(R^2 - r^2)$ for each section.

Intuitive explanation

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