Find the volume of the solid generated by revolving the...

Find the volume of the solid generated by revolving the region in the first quadrant bounded by the curve $y^{2}=x^{3}$, the line $x=4,$ and the $x$ -axis:
(a) about the line $x=4$
(b) about the line $y=8$

Key Concepts

Setting Up Integration Bounds

In problems of volumes by revolution, determining the correct limits of integration is essential. These bounds are established by the intersecting curves or lines that define the region being rotated. Accurately identifying the region's domain ensures that the integration captures the entire volume of the solid.

Cylindrical Shell Method

The cylindrical shell method provides an alternative by slicing the solid parallel to the axis of rotation, which produces thin cylindrical shells. The volume element is determined based on the circumference, height, and thickness of each shell, and integration is used to sum the contributions of all shells over the region. This method is particularly useful when the integration with respect to one variable simplifies the description of the region or when the other method results in complex expressions.

Axis of Rotation and Coordinate Shifts

This key concept involves understanding how the location of the axis of rotation influences the setup of the integral. When the axis of rotation is not one of the coordinate axes, the expressions for the distances (radii) from points in the region to the axis must be adjusted accordingly. Proper accounting for these distance shifts is critical in both the disc/washer and shell methods.

Disc/Washer Method

The disc or washer method is a technique used to compute the volume of a solid of revolution when the slices taken perpendicular to the axis of rotation form circles or annuli (washers). This method involves setting up integrals where the area of each disc or washer is computed based on the distance (radius) from the axis of rotation, and then integrating these areas over the appropriate bounds.

Volume of Solids of Revolution

This concept refers to the process of finding the volume of a three?dimensional solid generated by rotating a plane region around a given axis. The principle relies on the idea that the rotation of a two-dimensional area produces a solid whose volume can be determined by integrating the areas of cross-sectional slices that make up the solid.

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