Problem 42 - Thomas Calculus in SI Units

Question

Find the volume of the solid generated by revolving each region about the \(y\) -axis. The region enclosed by the triangle with vertices (0,1),(1,0) and ( 1,1 )

Step-by-Step Solution

Answer

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

Step 1: Understanding the Shape

The region enclosed by the triangle with vertices (0,1), (1,0), and (1,1) is a right triangle. The base is along the x-axis between x = 0 and x = 1, and the height is along the y-axis from y = 0 to y = 1.

Step 2: Identify the Function for Revolution

The line between the vertices (0,1) and (1,0) represents the hypotenuse of the triangle. The equation of this line can be found using the slope-intercept form: \( y = 1 - x \).

Step 3: Set up the Integral for the Volume

To find the volume created by revolving the triangle around the y-axis, we use the method of cylindrical shells. The formula for the volume of revolution using cylindrical shells is: \( V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx \). Here, \( f(x) = 1 - x \), \( a = 0 \), and \( b = 1 \).

Step 4: Calculate the Integral

Substitute \( f(x) \) and the limits of integration into the integral: \[V = 2\pi \int_{0}^{1} x(1-x) \, dx = 2\pi \int_{0}^{1} (x - x^2) \, dx\]Evaluate the integral: \[= 2\pi \left[ \frac{x^2}{2} - \frac{x^3}{3} \right]_{0}^{1}\]\[= 2\pi \left( \frac{1}{2} - \frac{1}{3} \right)\]\[= 2\pi \left( \frac{3}{6} - \frac{2}{6} \right) = 2\pi \cdot \frac{1}{6} \]\[= \frac{\pi}{3}\]

Step 5: Evaluate and Simplify

The result from Step 4 gives the volume of the solid. Simplifying the expression \( 2\pi \cdot \frac{1}{6} \) yields \( \frac{\pi}{3} \). This is the volume of the solid obtained by rotating the described triangle about the y-axis.

Key Concepts

Cylindrical Shells Method

When finding the volume of a solid of revolution, the cylindrical shells method is a powerful approach. It is particularly useful when the solid is generated by rotating a region around a vertical line (like the y-axis) and when the axis of rotation is parallel to the axis of integration.

In this method, you imagine the solid as being made up of thin, hollow cylinders (shells). The height of a shell corresponds to the function defining the boundary of the region, and the radius is the distance from the axis of rotation to the shell. The thickness of the shells is determined by the infinitesimally small change in the x-direction, denoted as \( dx \).

To determine the volume of a single shell, we use the formula for the lateral surface area of a cylinder (\( 2\pi \times \text{radius} \times \text{height} \)), and integrate this expression over the interval that encompasses the region of interest. This gives the total volume of the solid formed. This method is often preferred when dealing with the rotation about the y-axis due to its symmetry with the integral setup.

Integral Calculus

Integral calculus is a branch of mathematics that helps us calculate quantities like areas under curves, total accumulation, and volumes of solids. It is based on the concept of the integral, which is a limit of a sum of infinitesimal quantities known as 'infinitesimals.'

In the problem of finding the volume of a solid of revolution, integral calculus allows us to add up the volumes of infinitely small cylindrical layers to find the complete volume of the solid. This is done through the definite integral, which quantifies the total volume by evaluating the integral of the function that describes the shell's height times the distance to the axis over the given bounds.

Using the formula, \( V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx \), we find the volume by integrating the product of radius (\( x \)) and height (\( f(x) \)) across the limits \( a \) and \( b \). The integral symbol \( \int \) represents the process of summation, ensuring that we account for the entire volume without missing any parts of the solid.

Rotation about the y-axis

Rotation about the y-axis involves creating a three-dimensional solid by revolving a two-dimensional region around the vertical y-axis. This technique is a classic application of volume of revolution in calculus, where we take a known shape—like the triangle in this exercise—and sweep it around an axis to form a solid.
By revolving the triangle defined by the vertices (0,1), (1,0), and (1,1) around the y-axis, we essentially create a solid shape that is symmetrical about the y-axis. This results in the generation of a solid whose volume we seek to compute. Rotating around the y-axis, as opposed to the x-axis, influences how we set up our integral, impacting the variables we use and the method we choose to solve the problem.

This specific setup often leads us to the cylindrical shells method because it naturally aligns with integrating along parallel horizontal strips moving vertically, making it easy to address the structure and symmetry of the solid created by such rotation.