Problem 23 - Essential Calculus

Question

Use the shell method to find the volume of the solid generated by revolving the plane region about the given line. $$ y=4 x-x^{2}, \quad y=0, \text { about the line } x=5 $$

Step-by-Step Solution

Answer

The volume of the solid generated by revolving the plane region about the line \(x=5\) is \( \frac{128π}{5}\) cubic units.

Step 1: Define the radius and height of the cylindrical shells

The shell method says that the volume of a shell is approximately \(2π\) times the radius times the height times the thickness. Since we're revolving around the line \(x=5\), the radius of a shell is \((5-x)\), and the height is the value of the function, \(4x-x^2\). The thickness of the shell is a small change in x, or \(dx\).

Step 2: Write the integral to find the volume

According to the shell method, the volume of the solid is equal to the integral from a to b of \(2π\) times the radius times the height times the thickness of the shell. Here, we have \(2π\) integral from 0 to 4 of \((5-x)(4x-x^2)dx\).

Step 3: Compute the integral

Now we need to solve the integral. We do this by using the power rule for integration and the rules for integrating sum/difference of functions. After integrating, the volume is \( \frac{128π}{5}\).

Step 4: Evaluate the limits of integration

We then substitute the limits of the integration into the antiderivative, i.e., \( \frac{128π}{5} - 0\), to get the final volume.

Key Concepts

Solid of Revolution

A solid of revolution is a three-dimensional shape that is created when a two-dimensional area is revolved around a line (the axis of rotation). Imagine tracing a circular path with a line segment; the surface it creates would be a cylindrical shell, and the volume of that shell contributes to the total volume of the solid. Creating these solids is a common task in geometry and can also be an aesthetic endeavour in crafts like pottery where a clay design is rotated to form symmetrical vessels. In our exercise, revolving the plane area bounded by the curve of the function and the line 'about the line x=5' creates the solid of revolution we are interested in calculating.

Integral Calculus

Integral calculus is a branch of mathematics focused on the accumulation of quantities and the areas under and between curves. When it comes to finding volumes, we often use definite integrals, which give us a way to calculate the total accumulation of a quantity - in this case, volume - within certain bounds, a to b. Think of integration as a process of summing infinitesimally small slices to find the whole. The integral sign, which looks like a stretched 'S', symbolizes this summing process, and when paired with appropriate limits and a function, it enables us to find areas, volumes, and more.

Cylindrical Shells

Cylindrical shells are thin, hollow tubes with a circular cross-section used in the shell method for volume calculation. When you're calculating the volume of a solid of revolution using cylindrical shells, you see the solid as a stack of these tubes. Each shell's volume is determined by its radius (distance from the axis of rotation), its height (determined by the function), and its thickness (an infinitesimally small value, dx or dy, depending on the axis of rotation). By summing the volumes of an infinite number of such shells, we effectively fill the entire solid, giving us its total volume.

Volume by Integration

Calculating volume by integration is a powerful application of integral calculus. The shell method, specifically, finds the volume of a solid of revolution by integrating the volume of each cylindrical shell. In this case, the integral runs over the interval from a to b, where a and b represent the bounds of the region being revolved. By setting up an integral that multiplies 2π (the circumference of the shell's circular edge) with the radius, height, and thickness of each shell, we sum the volumes of an infinite number of infinitesimally thin shells—accurately calculating the solid's volume.