Problem 20 - Essential Calculus

Question

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the \(x\) -axis. $$ y=\sqrt{x+2}, \quad y=x, \quad y=0 $$

Step-by-Step Solution

Answer

The volume of the solid generated by revolving the region bounded by the curves \(y = \sqrt{x+2}, \quad y=x, \quad y=0\), about the \(x\)-axis is \(\frac{19\pi}{2}\)

Step 1: Identify the radius and height

For the shell method, the distance from the shell to the axis of rotation is our radius. Since we're rotating around the \(x\)-axis, our radius will be \(y\). The height of the shell will be the distance along the axis of rotation, which is \(x\). So here, \(y\) is the radial function and \(h = \sqrt{x+2} - x\) is the height function.

Step 2: Identify the bounds of integration

To find the bounds in the shell method, we find where the functions intersect. So we solve \( y = x\) and \(y = \sqrt{x+2}\). After solving these two equations, we get \(x=1\) and \(x=4\). So our bounds of integration are \(1\) and \(4\).

Step 3: Set up and solve the integral

Now we're ready to set up and solve the integral for the volume. So \(V = 2 \pi \int_{1}^{4} yh \, dx = 2\pi \int_{1}^{4}y(\sqrt{x+2} - x) \, dx\). After calculating this integral, we get \(V = 2\pi(\sqrt{2}[2\sqrt{2}+3]-\frac{9}{2}) = \frac{19\pi}{2}\).

Key Concepts

Volume of Revolution

When we talk about the volume of revolution, we mean the volume of a 3-dimensional object that we create by revolving a 2-dimensional region around an axis. Imagine spinning a flat shape around a line; the shape will form a solid object. In the given exercise, we have a region bounded by curves on a plane and we rotate it around the x-axis to form the 3D solid.

To find this volume, we use methods involving integration, with the shell method being a popular choice for certain problems. The method uses cylindrical shells with heights and radial distances defined by the functions that bound the region in the plane.

The shell method often gives us an easier understanding of the geometry involved. By viewing the solid as a sum of thousands of tiny cylindrical shells, we can calculate the overall volume efficiently.

Integration Bounds

Integration bounds define the limits of the region we're dealing with, denoting where the region starts and ends. For the shell method, these bounds impact where our shell exists along the axis of rotation, determining the range for our integral.

In the provided problem, we have two functions: \( y = x \) and \( y = \sqrt{x+2} \). To find where these equations intercept, we equate and solve these functions, resulting in points \( x = 1 \) and \( x = 4 \). These x-values become our bounds of integration.

• Lower Bound: 1
• Upper Bound: 4

These bounds tell us the interval within which we need to calculate, ensuring we're capturing the complete area rotated to form the solid.

Radial Function

The radial function represents the distance from any point in the region to the axis of rotation. This distance is crucial as it dictates the size of the cylindrical shell we use in our method. For the shell method around the x-axis, this radial distance is given by \( y \) itself.

In the problem at hand, our shell's radius is simply the y-value of each point within the region, transforming the region into a set of shells. In simpler terms, imagine measuring from the axis of rotation directly up to a point on our curve. This measurement, being variable, changes as we move along the region, impacting the shape and size of shells formed.

Understanding this distance helps set up the integral correctly, ensuring each shell is appropriately sized according to its position within the region.

Height Function

The height function in the shell method is the vertical distance between two curves within the region being revolved. This height is used to find the height of each cylindrical shell formed during revolution.

In the solution, the height function is \( h = \sqrt{x+2} - x \), capturing the difference in y-values between the upper curve \( y = \sqrt{x+2} \) and the lower line \( y = x \).

• Upper Curve: \( y = \sqrt{x+2} \)
• Lower Curve: \( y = x \)

This height function ensures the shells have the appropriate height across the region, forming the correct volume when rotated. Essentially, each shell height varies as we slide along the x-axis, accounting for the different shapes formed by the curves involved.