Question

FInd the volume of the solid generated when the region \(R\) bounded by the given curves is revolved about the indicated axis. Do this by performing the following steps. (a) Sketch the region \(R\). (b) Show a typical rectangular slice properly labeled. (c) Write a formula for the approximate volume of the shell generated by this slice. (d) Set up the corresponding integral. (e) Evaluate this integral. \(x=y^{2}, y=2, x=0 ;\) about the line \(y=2\)

Short answer

The volume is \( \frac{32\pi}{3} \).

Step-by-step solution

Sketch the Region R

The curves given are \(x = y^2\), \(y = 2\), and \(x = 0\). The region \(R\) is bounded by these curves in the \(xy\)-plane. To sketch, note that \(x = y^2\) is a parabola opening to the right, \(y = 2\) is a horizontal line, and \(x = 0\) is the \(y\)-axis. The intersection of these curves in the first quadrant forms region \(R\).

Show a Typical Rectangular Slice

A typical vertical slice within the region \(R\) is taken at a height \(y\) with width \(dy\). This slice is bounded by the curve \(x = y^2\) and the line \(x = 0\). The height of the rectangular slice is from \(0\) to \(y\), and it is revolved about the line \(y = 2\).

Write Approximate Volume of the Shell

Using the method of cylindrical shells, the height of the slice is \(y\), and the radius of the shell is \(2 - y\) (since it is revolved about \(y = 2\)). The thickness is \(dy\). The volume of a shell is approximately: \[ dV = 2\pi \cdot \text{(average radius)} \cdot \text{(height)} \cdot \text{(thickness)} = 2\pi (2-y)(y)\, dy \]

Set Up the Integral

Integrate the volume of the shells from \(y = 0\) to \(y = 2\): \[ V = \int_0^2 2\pi (2-y)(y)\, dy \]

Evaluate the Integral

Solve the integral to find the volume: \[V = \int_0^2 2\pi (2y - y^2)\, dy = 2\pi \left[ \frac{4y^2}{2} - \frac{y^3}{3} \right]_0^2 \]\[= 2\pi \left(2(4) - \frac{8}{3}\right) = 2\pi \left(8 - \frac{8}{3}\right) \]\[= 2\pi \left(\frac{24}{3} - \frac{8}{3}\right) = 2\pi \left(\frac{16}{3}\right) = \frac{32\pi}{3} \]

Key concepts

Cylindrical Shell Method

The cylindrical shell method is a powerful technique used to find the volume of solids of revolution. Instead of using disks or washers, we slice the solid into cylindrical shells.
This process involves integrating the volume of these thin shells as they revolve around an axis. When visualizing this, imagine the solid as a collection of nested cylindrical wrappers. Each wrapper is described by a radius, a height, and a thickness. It's like peeling a roll of paper, layer by layer. The formula for the volume of a cylindrical shell is given by:

1. Radius ((a)): This is the distance from the axis of rotation to the slice, which can be described for each shell based on its position.
2. Height ((h)): This typically comes from the difference in the values of a function over a range.
3. Thickness: Represented as an infinitesimally small value ((dy)).

We calculate the volume of the shells by using the integral \[ dV = 2\pi \times (\text{radius}) \times (\text{height}) \times (\text{thickness}) \].
Integrating over the adequate limits will give the total volume of the solid.

Definite Integrals

Definite integrals play a vital role in calculating the exact volume in the cylindrical shell method. They allow us to sum up an infinite number of infinitesimally small volumes, providing an exact result. In our context, the definite integral \[ \int_a^b f(x) \, dx \] describes the accumulation of the volume of shells between specific bounds that correspond to the region's limits.
The bounds, (a) and (b), represent the interval over which we want to integrate, and the function (f(x)) determines the changing features of our shells. When setting up a definite integral for this method:

• Decide on the variable of integration (typically depending on the axis of revolution).
• Determine the limits of integration, derived from the intersection points of the curves describing the region.
• Focus on whether the radius and height functions need constants or adjustments, especially when rotating about a line that is not an axis.

Definite integrals give a concrete value that represents the volume we are interested in. They are crucial for confirming your results.

Integration Techniques

To evaluate the volume using definite integrals, it's often necessary to employ different integration techniques. These methods help simplify and solve integrals that may initially appear complex. Here are some common techniques:

• Simplification: Break down the expression into standard integral forms so that basic antiderivatives can be used.
• Polynomial Expansion: Expanding terms like \((2-y)y\) helps simplify the integration process, making it straightforward to apply basic rules.
• Substitution: When the integral is complicated, substitution can make it easier by changing variables to simplify the integrand.

When integrating the definite integral for cylindrical shells, apply these steps methodically. Combine your work into a simplified expression before integrating. Each step enhances comprehension and ensures accuracy. Through thoughtful application of these techniques, you can smoothly evaluate the integral and obtain the correct volume.

Sketching Regions in Calculus

Sketching regions is often the initial step when solving problems involving volumes of revolution. It provides a visual framework that helps you understand which part of the plane you are revolving. To sketch regions effectively:

• Identify and carefully draw all involved curves on the coordinate plane, making note of key intercepts and intersections.
• Shade the area of interest—this is the region that will be revolved. Be clear about your axis of revolution.
• Include a representative slice, which you will use to build up the volume. This slice is typically perpendicular to the axis of revolution.

Sketching is crucial, as it helps you:

• Determine the limits of integration by identifying the endpoints of the region.
• Visualize how the rotation affects the shape and size of the resulting solid.

A good sketch underpins a correct setup of the problem and aids in visualizing the geometry of the solid being formed. Mastering this step can significantly ease the subsequent calculations.