Problem 30 - Thomas Calculus in SI Units

Question

For some regions, both the washer and shell methods work well for the solid generated by revolving the region about the coordinate axes, but this is not always the case. When a region is revolved about the \(y\) -axis, for example, and washers are used, we must integrate with respect to \(y .\) It may not be possible, however, to express the integrand in terms of \(y .\) In such a case, the shell method allows us to integrate with respect to \(x\) instead. Compute the volume of the solid generated by revolving the triangular region bounded by the lines \(2 y=x+4, y=x,\) and \(x=0\) about a. the \(x\) -axis using the washer method. b. the \(y\) -axis using the shell method. c. the line \(x=4\) using the shell method. d. the line \(y=8\) using the washer method.

Step-by-Step Solution

Answer

a) Use washer method to integrate with respect to x from 0 to 4. b) Use shell method to integrate from 0 to 4. c) Use shell method from 0 to 4 around x=4. d) Re-express functions in terms of y and use washer method on [0, 4].

Step 1: Identify the Equations of the Lines

The given region is bounded by the lines \(2y = x + 4\), \(y = x\), and \(x = 0\). Simplify these equations to express them in forms of \(y\) and \(x\). The line \(2y = x + 4\) can be rewritten as \(y = \frac{x}{2} + 2\). The region is in the first quadrant where \(x \geq 0\).

Step 2: Find Intersection Points

Find the intersection points of the lines to determine the region's vertices. Solve \(y = x\) and \(y = \frac{x}{2} + 2\) to find their intersection by setting them equal: \(x = \frac{x}{2} + 2\). Solving gives \(x=4\), thus \((4, 4)\) is one intersection point. The points of intersection are \((0, 0)\), \((4, 4)\), and by substituting \(x=0\) in both equations, we confirm \((0, 2)\) for \(y = \frac{x}{2} + 2\) and \((0, 0)\) for \(y = x\).

Step 3: Volume Using the Washer Method about the x-axis

The formula for volume using the washer method about the \(x\)-axis is \( V = \pi \int_a^b \left( [f(x)]^2 - [g(x)]^2 \right)\, dx \). Here, \(f(x) = \frac{x}{2} + 2\) and \(g(x) = x\), from \(x = 0\) to \(x = 4\). Integrate \(\pi \int_0^4 \left( \left( \frac{x}{2} + 2 \right)^2 - x^2 \right)\, dx\). Computing the integral provides the volume.

Step 4: Volume Using the Shell Method about the y-axis

The formula for volume using the shell method about the \(y\)-axis is \( V = 2\pi \int_a^b (x)(f(x) - g(x))\, dx \). The limits for \(x\) are from \(0\) to \(4\). Thus, integrate \(2\pi \int_0^4 x(x - (\frac{x}{2}+2))\, dx = 2\pi \int_0^4 x(x - \frac{x}{2} - 2)\, dx \). Solving this integral calculates the volume.

Step 5: Volume Using the Shell Method about the Line x = 4

For revolving about \(x = 4\), use the shell method: \( V = 2\pi \int_a^b (4-x)(f(x) - g(x))\) within the integration limits from \(x=0\) to \(x=4\). Therefore, use \( V = 2\pi \int_0^4 (4-x)(\frac{x}{2}+2-x)\, dx\). Calculate this integral to find the volume.

Step 6: Volume Using the Washer Method about the Line y = 8

When using the washer method around \(y = 8\), the formula is \(V = \pi \int_a^b \left( [R(y)]^2 - [r(y)]^2 \right) \, dy \). Here, rearrange the equations to functions in terms of \(y\): \(y = x\) as \(x = y\) and \(y = \frac{x}{2} + 2\) to \(x = 2y - 4\). The interval is from \(y = 0\) to \(y = 4\). The outer radius \(R(y)\) is \(8 - (2y-4)\) and inner radius \(r(y) = 8-y\). Integrate \(\pi \int_0^4 [(8-(2y-4))^2 - (8-y)^2] \, dy\) to find the volume.

Key Concepts

Washer Method

The Washer Method is a popular technique in calculus used to find the volume of a solid formed by revolving a region around a given axis. Imagine the shape as a stack of actual washers or discs, where the solid's volume is the sum of these washers' volumes.

To apply the Washer Method, identify the outer radius and the inner radius of the washers. The outer radius represents the distance from the axis of revolution to the outer edge of the washer, while the inner radius is to the inner edge. For a washer revolving around the x-axis, the volume is calculated using the integral:

• \[ V = \pi \int_a^b \left( [R(x)]^2 - [r(x)]^2 \right)\, dx \]

Here, \(R(x)\) and \(r(x)\) are the functions describing the outer and inner radii with respect to x. The limits of integration, \(a\) and \(b\), are determined by the span of the region along x.

In the original exercise, the washer method was used by identifying \(f(x) = \frac{x}{2} + 2\) and \(g(x) = x\) with respect to the x-axis from \(x=0\) to \(x=4\). This setup allows for calculating the volume of the solid using a straightforward integral.

Shell Method

In calculus, the Shell Method provides an elegant way to compute the volume of a solid of revolution by slicing the solid into cylindrical shells. This method is particularly useful when the washer method becomes too complicated or impossible to use due to alignment constraints.

The formula for the Shell Method, when revolving around the y-axis, is:

• \[ V = 2\pi \int_a^b x(f(x) - g(x))\, dx \]

Here, \(x\) is the distance from the y-axis to the shell, and \(f(x)\) and \(g(x)\) represent the height of the outer and inner functions respectively.

In the exercise, the Shell Method is used for two distinct revolving scenarios:

• Revolving around the y-axis, where the function \(f(x) - g(x)\) results in the volume calculation as the region bounded by lines \(y = x\) and \(y = \frac{x}{2}+2\). The limits from \(x=0\) to \(x=4\) ensure the correct volume is determined.
• Revolving about the line \(x=4\), this involves adapting the method to account for horizontal shifting.

Revolving Solids

Revolving Solids is a general concept in calculus involving creating a three-dimensional shape by rotating a two-dimensional region around a fixed line, commonly known as an axis of revolution.

Common axes include the x-axis, y-axis, or lines parallel to these, such as \(x=4\) or \(y=8\), as seen in the given exercise. When a region is revolved, the shape of the solid and the method of integration are affected by the axis choice. Hence, it's important to understand both shell and washer methods to choose the simplest approach for solving volume integrals.

Each axis of revolution might require manipulating the initial equations to fit the desired method: moving from functions of \(x\) to \(y\) or vice versa, depending on what's simplest to integrate. The techniques ensure that the integral accounts for all volume between the defined boundary lines, leading to a precise calculation of the solid's size.

Integration Techniques

Understanding integration techniques is crucial in applying both the Washer and Shell methods in calculus. Integration allows us to calculate the "sum" of infinitesimally small elements to find the whole volume.

When dealing with the Washer and Shell methods:

• Identify the correct limits of integration which correspond to the region's boundaries.
• Set up the integrals using the respective formulas and double-check whether you integrate with respect to \(x\) or \(y\).
• Solve the integrals using standard calculus techniques, which might include substitution or integration by parts if the integral is complex.

For instance, in the original solutions, applying integration involved clarifying bounds and expressions of the given functions as either \(x\) or \(y\), based on the axis of revolution. Consistently practicing these techniques allows for tackling even complex volume problems with confidence.