Question

A region of the Cartesian plane is described. Use the Shell Method to find the volume of the solid of revolution formed by rotating the region about each of the given axes. Region bounded by \(y=2 x, y=x\) and \(x=2\). Rotate about: (a) the \(y\) -axis (b) \(x=2\) (c) the \(x\) -axis (d) \(y=4\)

Short answer

(a) \( \frac{16\pi}{3} \), (b) \( \frac{16\pi}{3} \), (c) \( \frac{15\pi}{2} \), (d) \( \frac{15\pi}{2} \).

Step-by-step solution

Identify the Region

The region is bounded by the lines \( y = 2x \), \( y = x \), and \( x = 2 \). Sketch these lines on the Cartesian plane to visualize the area of interest.

Set Up Integral for Part (a): Rotation About the y-axis

Use the shell method formula for rotation about the \( y \)-axis: \int_{a}^{b} 2\pi x (f(x) - g(x)) \, dx. Here, \( f(x) = 2x \) and \( g(x) = x \). The integral becomes \int_{0}^{2} 2\pi x ((2x) - x) \, dx.

Evaluate Integral for Part (a)

Simplify the expression to get \int_{0}^{2} 2\pi x (x) \, dx = \int_{0}^{2} 2\pi x^2 \, dx. Calculate this integral to find the volume: \[ 2\pi \int_{0}^{2} x^2 \, dx = 2\pi \left[ \frac{x^3}{3} \right]_{0}^{2} = 2\pi \left( \frac{8}{3} - 0 \right) = \frac{16\pi}{3}. \]

Set Up Integral for Part (b): Rotation About x = 2

Shift the axis of rotation to \( x = 2 \). Use the shell method formula: \int_{a}^{b} 2\pi (R - x) (f(x) - g(x)) \, dx where \( R = 2 \). The integral is \int_{0}^{2} 2\pi (2 - x)(2x - x) \, dx.

Evaluate Integral for Part (b)

Simplify the expression to get \int_{0}^{2} 2\pi (2 - x)x \, dx = \int_{0}^{2} 2\pi (2x - x^2) \, dx. Calculate the integral: \[ 2\pi \left[ \left( x^2 - \frac{x^3}{3} \right) \right]_{0}^{2} = 2\pi \left( 4 - \frac{8}{3} \right) = \frac{16\pi}{3}. \]

Set Up Integral for Part (c): Rotation About the x-axis

Switch to rotating around the \( x \)-axis. Use the formula for cylindrical shells: \[ V = \int_{a}^{b} 2\pi y (x(y) - x(y)) \, dy. \] Rewrite the equations in terms of \( y \): \( x = \frac{y}{2}, x = y \), hence the integral becomes \[ \int_{0}^{2} 2\pi y \, dy. \]

Evaluate Integral for Part (c)

The integral simplifies to \[ \int_{1}^{4} 2\pi \left( \frac{y}{2} \right) \, dy = \pi \int_{1}^{4} y \, dy. \] Calculate this integral: \[ \pi \left[ \frac{y^2}{2} \right]_{1}^{4} = \pi (8 - 0.5) = \frac{15\pi}{2}. \]

Set Up Integral for Part (d): Rotation About y = 4

Set up the integral with the axis of rotation at \( y = 4 \). Use the shell formula: \[ \int_{0}^{2} 2\pi (4 - y)(x(y) - x(y)) \, dy. \] Express \( x \) in terms of \( y \): \( x= \frac{y}{2}, x = y \) resulting in \[ \int_{1}^{4} 2\pi (4-y)(\frac{y}{2} - y) \, dy. \]

Evaluate Integral for Part (d)

Simplify to get \[ 2\pi \int_{1}^{4} (4-y)(\frac{y}{2} - y) \, dy = 2\pi \int_{1}^{4} \left( 2y - y^2 \right) \, dy. \] Calculate this integral: \[ 2\pi \left[ ( \frac{y^2}{2} - \frac{y^3}{3} ) \right]_{1}^{4} = 2\pi \left( \frac{8}{3} - 0.5 \right) = \frac{15\pi}{2}. \]

Key concepts

Shell Method

The Shell Method is a technique in integral calculus used to find the volume of a solid of revolution. It is particularly useful when the region is rotated around an axis different from the axis of the function. This method uses cylindrical shells to calculate volumes.
The basic formula for the Shell Method when rotating about the y-axis is:

• \( V = \int_{a}^{b} 2\pi x (f(x) - g(x)) \, dx \)

Here, \(x\) represents the distance from the axis of rotation, and the height of the shell is \(f(x) - g(x)\). The limits \(a\) and \(b\) indicate the points where the shell starts and ends.
By slicing the region into thin vertical strips parallel to the axis of rotation, the volume of each infinitesimally thin shell is calculated and summed using integration.

Integral Calculus

Integral calculus is a branch of mathematics focused on accumulation values and areas under or between curves. It is essential for finding volumes when you deal with three-dimensional shapes, particularly in the context of solids of revolution.
In the context of finding the volume of revolution using the Shell Method, the area function is described by integrals like:

• \( \int_{a}^{b} 2\pi x (f(x) - g(x)) \, dx \)

The integration effectively adds up an infinite number of infinitesimally thin cylindrical shells to build the entire volume.

Cartesian Plane

The Cartesian Plane is a two-dimensional coordinate system defined by a vertical line called the y-axis and a horizontal line called the x-axis. It provides a way to graph mathematical functions like \(y = 2x\) and \(y = x\), as mentioned in the problem.
The use of the Cartesian plane allows us to easily visualize and sketch the bounded region. In this exercise, plotting the lines helps in identifying the region to be rotated for forming the solid of revolution. Consequently, these visualizations aid in setting up the correct integral boundaries for calculating volumes.

Axis of Rotation

The axis of rotation is a line around which a two-dimensional region rotates to form a three-dimensional solid. In our context, this could be the x-axis, y-axis, or any vertical or horizontal line, like x = 2 or y = 4.
The choice of the axis of rotation greatly influences the integral setup when using the Shell Method.

• Rotation about the y-axis requires shells with the radius equal to the x-coordinate value.
• Rotation about x = 2 shifts the axis away from the y-axis, requiring adjustments in the radius calculation: \( R - x \).
• Rotation about the x-axis involves transforming equations into \( y \) terms and adjusting the setup accordingly.

Understanding how the axis changes the setup is crucial to correctly applying the Shell Method formulas for volume calculation.