Question

In Exercises 23-28, find the volume of the solid generated by revolving the region about the y-axis. the region in the first quadrant bounded above by the parabola \(y=x^{2},\) below by the \(x\) -axis, and on the right by the line \(x=2\)

Short answer

The volume of the solid generated is \(8\pi\) cubic units.

Step-by-step solution

Identify the boundaries and limits

The volume of the solid is bounded above by the parabola \(y=x^{2}\), below by the \(x\)-axis, and on the right by the line \(x=2\). So, the lower limit of integration is 0 and the upper limit is 4 (as \(y=2^{2}\) ).

Express \(x\) as a function of \(y\)

The horizontal radius from the \(y\)-axis to the curve \(y=x^{2}\) is \(x=\sqrt{y}\). The vertical thickness of the disk is \(\delta y\). The area of the disk is \(\pi (radius)^{2}\) which in this case is \(\pi (\sqrt{y})^{2}=\pi y\). Therefore the differential volume element is \(\pi y \delta y\).

Integrate to find the volume

The volume of the solid is obtained by integrating the differential volume element from y = 0 to y = 4. Therefore, \(V = \int_{0}^{4} \pi y \delta y = \pi \int_{0}^{4} y dy\). Calculating the integral gives \(V = \frac{1}{2}\pi[ y^{2}]_{0}^{4} = 8\pi\).

Key concepts

Volume by Integration

Understanding the concept of calculating volume through integration is crucial when dealing with solids of revolution. The volume of such a solid can be determined by summing an infinite number of infinitesimally thin cross-sections. Imagine slicing the solid parallel to the axis of rotation and sum each slice's volume to determine the whole.

The process begins with identifying the shape of the cross-section and its area in terms of a single variable. In the given example, we revolve the region bounded by a parabola around the y-axis, leading to circular cross-sections perpendicular to the axis of rotation.

Once the area’s expression is determined, it is integrated across the interval corresponding to the solid's limits. This integral effectively 'adds up' the volumes of each disk, resulting in the total volume of the solid. The integration limits are derived from the bounds of the region being rotated and must be carefully identified, as in the visualization step provided in the problem.

Disk Method

The disk method is a specific application of volume by integration suited for solids of revolution when a region is rotated around an axis. Imagine taking a shape, like a slice of pie, and spinning it around a stick running through the center—a solid with circular cross-sections is formed.

This method involves visualizing the solid as an assembly of disks, each with a radius measured from the axis of rotation to the function defining the boundary, and a thickness that is infinitesimally small, represented by \( \text{d}y \) or \( \text{d}x \), depending on the axis of rotation.

In our given example, the radius is the distance from the y-axis to the parabola \(x = \sqrt{y}\). The area of each disk is therefore \( \pi y \) (\(y\) being the function squared). Integrating these areas from the lower to the upper bound as determined in the first step of the solution provides the solid's volume. The integration bounds align with the limits of the revolving region, which are established by the intersection points of the function with the axis of rotation or the boundary lines.

Volumes of Solids with Known Cross Sections

Calculating the volume of solids with known cross sections involves understanding the specific shape of the cross-section perpendicular to the axis of rotation. Each cross-section contributes a 'slice' to the total volume.

In the case of our exercise, rotation about the y-axis leads to circular cross-sections—discs. For other shapes, the area formula varies (e.g., squares, rectangles, or triangles), which must be integrated along the axis of revolution. Knowing the shape of the cross-section is pivotal because its area is the function integrated to determine the volume. This concept can be applied to any solid where the cross-sectional area can be expressed as a function of either x or y.

In practice, this concept allows for a wide variety of problems to be approached in a systematic way. Regardless of the complexities of the shape of the rotating region or the cross-section, if the area can be written mathematically in terms of the axis of revolution, integration can yield the volume of the resulting solid. Following the disk method, as shown in the exercise, provides a framework to dissect such problems and arrive at a solution through integration.