Problem 28 - Thomas Calculus in SI Units

Question

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves about the \(y\) -axis. The region enclosed by \(x=y^{3 / 2}, \quad x=0, \quad y=2\)

Step-by-Step Solution

Answer

The volume is \(4\pi\) cubic units.

Step 1: Identify the Bounded Region

The given curves and lines are \(x = y^{\frac{3}{2}}\), \(x = 0\), and \(y = 2\). These outlines describe a region in the \(xy\)-plane. \(x = y^{\frac{3}{2}}\) is a curve that bounds from the right, \(x = 0\) bounds from the left (the \(y\)-axis), and \(y = 2\) is a horizontal line marking an upper limit. The region of interest is where these curves bound together.

Step 2: Apply the Disk Method

To find the volume generated by revolving this region about the \(y\)-axis, we use the disk method. The formula is: \[ V = \pi \int_{a}^{b} [R(y)]^2 \, dy \]where \(R(y)\) is the radius of revolution for each disk, equivalent to the \(x\)-coordinate of the curve \(x = y^{\frac{3}{2}}\) as it revolves around the \(y\)-axis. Thus, \(R(y) = y^{\frac{3}{2}}\).

Step 3: Set the Limits of Integration

The limits of integration correspond to the \(y\)-values that bound the region, given as \(y = 0\) and \(y = 2\) since \(x\) values start at \(x = 0\) and go up to \(x = y^{\frac{3}{2}}\). So the integral is set from \(0\) to \(2\).

Step 4: Set Up the Integral

Substitute \(R(y)\) into the volume formula:\[ V = \pi \int_{0}^{2} (y^{\frac{3}{2}})^2 \, dy \]Simplifying gives:\[ V = \pi \int_{0}^{2} y^3 \, dy \]

Step 5: Evaluate the Integral

Calculate the integral:\[ V = \pi \left[ \frac{y^4}{4} \right]_{0}^{2} \]Substituting the limits:\[ V = \pi \left( \frac{2^4}{4} - \frac{0^4}{4} \right) \]This simplifies to:\[ V = \pi \left( \frac{16}{4} \right) = 4\pi \]

Step 6: Conclusion

The volume of the solid formed by revolving the given region around the \(y\)-axis is \(4\pi\) cubic units.

Key Concepts

Volume of Solids of Revolution

The volume of solids of revolution is a fascinating concept in calculus. It involves creating a 3-dimensional solid by revolving a 2-dimensional region around an axis—commonly the x-axis or y-axis. This process transforms flat shapes into full-bodied figures, through methods like the **disk method** or **shell method**.

In our exercise, we use the disk method to determine the volume. When revolving around the y-axis, imagine stacking many thin circular disks along this axis. Each disk's volume contributes to the overall volume of the solid.

The formula for calculating volume using the disk method when rotating around the y-axis is:

• \[ V = \,\pi \, \int_{a}^{b} [R(y)]^2 \,dy \]

where \(R(y)\) is the distance from the axis of rotation to the curve, acting as the radius of each disk across the integration range from \(a\) to \(b\). Understanding this foundation helps us to transform flat sketches into rounded 3D shapes.

Integration

Integration plays a critical role in determining the volume of solids of revolution. It is the mathematical tool we use to add up the infinite number of tiny disks or shells to get the whole solid's volume.

To integrate successfully, you need to properly set up the integral. This involves identifying the function and the limits of integration, which are the bounds of the region being revolved. In this exercise, the function \( y^{\frac{3}{2}} \) describes the curve, and the limits of integration are from \( y=0 \) to \( y=2 \).

When integrating \( y^3 \), as derived in our example, you apply the power rule for integrals:

• \[ \int y^n \, dy = \frac{y^{n+1}}{n+1} + C \]

where \( C \) is the constant of integration. However, between definite limits, \( C \) cancels out. So understanding integration will give you the tools to piece together individual slices of the solid into one complete volume.

Bounded Region

A bounded region is essentially the area enclosed by curves and lines on a coordinate plane. In our specific problem, the curves \( x = y^{\frac{3}{2}} \), \( x = 0 \) (the y-axis), and \( y = 2 \) describe such an area.

Understanding this concept is crucial because it determines what part of the plane will be revolved to create a 3D solid. Each boundary plays a role:

• \( x = y^{\frac{3}{2}} \) forms a curving outer limit.
• \( x = 0 \) acts as a vertical boundary at the y-axis.
• \( y = 2 \) represents a horizontal cap.

Grasping the notion of bounded regions allows you to visualize the 2D sections that combine to create the whole structure once they are rotated around an axis.

Revolving Around the Y-Axis

Revolving around the y-axis is a type of rotation transformation where the bounded region spins around this vertical axis. It transforms a two-dimensional figure into a three-dimensional solid.

Visualize this process: imagine tracing the curve \( x = y^{\frac{3}{2}} \) along the y-axis, like twirling a ribbon until it forms a shell around the axis. The y-axis becomes a spine, and each point on the curve traces a circle, creating cylindrical disks.

The distance from each point on the curve to the axis defines the radius of these circles, crucial for calculating the volume. This procedure illustrates how rotation changes dimensions and outlines, emphasizing the importance of setting up integration correctly to reflect the radius at every point on the curve.