Problem 30 - Calculus Early Transcendentals

Question

Let \(R\) be the region bounded by the following curves. Use the washer method to find the volume of the solid generated when \(R\) is revolved about the \(x\) -axis. $$y=x, y=x+2, x=0, x=4$$

Step-by-Step Solution

Answer

Question: Determine the volume of the solid generated by revolving the region R, bounded by the curves \(y=x\), \(y=x+2\), \(x=0\), and \(x=4\), around the x-axis. Answer: The volume of the solid generated when the region R is revolved about the \(x\)-axis is \(V = \frac{104\pi}{3}\).

Step 1: Understand the given functions

The given functions are \(y=x\), \(y=x+2\), \(x=0\), and \(x=4\). The first two lines are oblique lines and the last two lines are vertical lines, which will form a parallelogram shape. Determine the points of intersection of all the functions to get the range for integration.

Step 2: Find the points of intersection

Set the two functions equal to each other and solve for x. You will get \(x = x+2 \Rightarrow x=2\). The points of intersection are \((0,0),(0,2),(4,4),\) and \((4,6)\), from which we see the range for integration is from \(x=0\) to \(x=4\).

Step 3: Determine inner and outer radii for the washer method

As the region R is revolved around the \(x\)-axis, we need to find the radius of the inner and outer circles of the washers formed. They are equal to the distance of the curves from the x-axis. The outer radius is the distance from the \(x\)-axis to \(y=x+2\), which is just given by \(x+2\). The inner radius is the distance from the \(x\)-axis to \(y=x\), which is just given by \(x\).

Step 4: Set up the integration formula

Using the washer method, the volume formula is given by: $$V = \pi \int_{a}^{b} [R_o^2(x) - R_i^2(x)] dx$$ Where \(R_o(x)\) is the outer radius function, \(R_i(x)\) is the inner radius function, and a and b are the limits of integration.

Step 5: Substitute the values and integrate

For our problem, the outer radius is \((x+2)^2\) and the inner radius is \(x^2\). The volume V integral will be as follows: $$V = \pi \int_{0}^{4} [(x+2)^2 - x^2] dx$$ Integrate, $$V = \pi \Biggr[\frac{1}{3}(x+2)^3 - \frac{1}{3}x^3 \Biggr]_0^4$$

Step 6: Calculate the volume

Finally, substitute the upper and lower limits and calculate the volume: $$V = \pi \Biggr[\frac{1}{3}(6)^3 - \frac{1}{3}(4)^3 - \frac{1}{3}(2)^3 \Biggr]$$ $$V = \frac{104\pi}{3}$$ Therefore, the volume of the solid generated when the region R is revolved about the \(x\)-axis is \(V = \frac{104\pi}{3}\).

Key Concepts

Volume of Revolution

When we talk about the volume of a solid of revolution, we are referring to the three-dimensional shape created by rotating a two-dimensional region around an axis. This concept is particularly useful in disciplines like engineering and physics, where such shapes frequently appear. To understand this, imagine taking a flat drawing of a region and spinning it around a line (the axis of revolution) -- the space that this spinning region occupies is your solid of revolution.
In our specific problem, we're revolving the region bounded by the lines \(y=x\), \(y=x+2\), \(x=0\), and \(x=4\) around the \(x\)-axis. This action creates a series of circular disks or "washers" stacked along the axis, producing a solid shape like a cylinder with a varying radius along its length. This is where the washer method comes into play, helping us determine the shape's overall volume.

Integral Calculus

Integral calculus is the branch of mathematics that deals with the summation of parts to find whole quantities. It's a tool used to calculate areas, volumes, and other concepts that can be summed into a total. When calculating the volume of a solid of revolution, integral calculus plays a critical role through techniques like the washer method.

In the washer method, the idea is to integrate across the region's boundaries, taking into account the shapes formed when the area is revolved around the axis. This method involves setting up an integral with specific limits from one bound to the other and subtracting the area of the inner disk from the area of the outer disk, both squared and expressed as functions of \(x\). The process of setting the definite integral to calculate the volume is key and relies on evaluating the integral of the difference between these outer and inner circle areas throughout the entire defined region.

Solid Geometry

Solid geometry is the study of 3D shapes, much like the ones formed through volumes of revolution. It involves understanding geometric figures in three-dimensional space—such as spheres, cubes, and the kinds of varying shapes we encounter using the washer method.
In this exercise, solid geometry helps us visualize and comprehend the final result of revolving a plane region around a line. The resulting shape is a hybrid of familiar geometric shapes where the dimensions are described by functions such as \(y=x\) and \(y=x+2\) that define the contours of the sphere as it revolved around the \(x\)-axis. Understanding these shapes’ dimensions helps us effectively set up our equations in integral calculus, refining the integration process as we calculated a specific solid's volume. This interplay between equations and geometric visualizations is what makes solid geometry a vital part of solving real-world problems in mathematics and its applications.