Question

Area and volume The region R is bounded by the curves x = y²+2,y=x−4, and y=0 (see figure).
b. Write a single integral that gives the volume of the solid generated when R is revolved about the x-axis.

Accepted Answer

First, identify the region R bounded by the curves: $$ x = y^2 + 2 $$, $$ y = x - 4 $$, and $$ y = 0 . $$Sketching or visualizing these curves helps understand the limits of integration and the shape of the region. Since the solid is generated by revolving the region R about the x-axis, consider using the method of cylindrical shells or washers/disks. Here, using the washer method with respect to $$ y  is $$convenient because the boundaries are given in terms of $$ y $$ and $$ x . $$Express the volume element as a washer with inner radius and outer radius measured from the x-axis. The outer radius is the distance from the x-axis to the upper curve, and the inner radius is the distance from the x-axis to the lower curve within the region. Since $$ y = 0  is $$one boundary, the radius will be $$ y $$ itself. Determine the limits of integration for $$ y  by $$finding the intersection points of the curves in terms of $$ y . $$Solve for the values of $$ y $$ where the curves intersect to establish the bounds of integration. Set up the integral for the volume using the washer method formula: $$ V = \pi \int_{a}^{b} \left( R_{outer}(y)^2 - R_{inner}(y)^2 \right) \, dy $$, where $$ R_{outer}(y) $$ and $$ R_{inner}(y) $$ are the outer and inner radii of the washers. Express these radii in terms of $$ y $$ using the given curves.

Area and volume The region R is bounded by the curves x = y²+2,y=x−4, and y=0 (see figure).
b. Write a single integral that gives the volume of the solid generated when R is revolved about the x-axis.

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Area and volume The region R is bounded by the curves x = y²+2,y=x−4, and y=0 (see figure).b. Write a single integral that gives the volume of the solid generated when R is revolved about the x-axis.

Verified video answer for a similar problem:

Key Concepts

Region Bounded by Curves

Volume of Solids of Revolution

Setting up a Single Integral for Volume

Area and volume The region R is bounded by the curves x = y²+2,y=x−4, and y=0 (see figure).
b. Write a single integral that gives the volume of the solid generated when R is revolved about the x-axis.