Problem 281 - Calculus Volume 1

Question

For the following exercises, use the theorem of Pappus to determine the volume of the shape. Rotating \(y=m x\) around the \(y\) -axis between \(x=0\) and \(x=1\)

Step-by-Step Solution

Answer

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

Step 1: Understand Pappus' Theorem

Pappus' Theorem states that the volume of a solid of revolution generated by rotating a plane figure around an external axis is equal to the product of the area of the figure and the distance traveled by its centroid. Here, we need to find the area of the triangle under the line between \(x=0\) and \(x=1\) and the distance traveled by its centroid around the \(y\)-axis.

Step 2: Determine the Area of the Triangle

The line \(y=mx\) forms a right triangle with the \(x\)-axis between \(x=0\) and \(x=1\). The height of the triangle is \(m\) at \(x=1\). The base is 1 unit (from \(x=0\) to \(x=1\)). Thus, the area \(A\) of the triangle is calculated as \[ A = \frac{1}{2} \times ext{base} \times ext{height} = \frac{1}{2} \times 1 \times m = \frac{m}{2}. \]

Step 3: Find the Coordinates of the Centroid

The centroid \((x_c, y_c)\) of a right triangle can be found using the formula \(x_c = \frac{b}{3}, y_c = \frac{h}{3}\), where \(b\) is the base and \(h\) is the height. Here, \(b=1\) and \(h=m\), so \[ x_c = \frac{1}{3}, \quad y_c = \frac{m}{3}. \]

Step 4: Calculate the Centroid's Distance Traveled

When the shape is rotated around the \(y\)-axis, the centroid at \((x_c, y_c) = (\frac{1}{3}, \frac{m}{3})\) travels in a circle. The radius \(r\) of that circle is \(x_c = \frac{1}{3}\). Therefore, the distance traveled by the centroid is the circumference of that circle: \[ ext{Distance} = 2 \pi r = 2 \pi \times \frac{1}{3} = \frac{2\pi}{3}. \]

Step 5: Apply Pappus' Theorem to Find the Volume

Using Pappus' Theorem, the volume \(V\) of the solid is the product of the area of the triangle and the distance traveled by its centroid: \[ V = A \times ext{Distance} = \frac{m}{2} \times \frac{2\pi}{3} = \frac{m \pi}{3}. \]

Step 6: Conclusion

The calculated volume of the solid of revolution, by rotating \(y=mx\) around the \(y\)-axis from \(x=0\) to \(x=1\), is \(\frac{m \pi}{3}.\)

Key Concepts

Pappus' Theorem

Pappus' Theorem is a powerful tool for calculating the volume of a solid of revolution. It is particularly useful when rotating a plane figure around an external axis. Let's break down the theorem in simple terms:

• The theorem states that if you have a flat figure and you rotate it around an external line (not through it), the resulting volume is found by multiplying the area of the original figure by the path traveled by its centroid.
• This external line acts as the axis of rotation. The centroid's path forms a circular track, and the length of this track is key to finding the volume.
• This means you don't have to individually calculate the volume of every small slice of the rotated figure, which saves a lot of computation time and effort.

To apply Pappus' Theorem effectively, you first find the area of the figure. Then, determine the distance traveled by the centroid during rotation. Multiply these two values, and you have your volume! In situations like this exercise, we see the theorem neatly simplifies the task.

Centroid

The centroid is like the 'center of mass' or 'balance point' of a shape. For practical purposes in geometric problems, understanding the concept of a centroid is essential.

• For a triangle, the centroid can be found using fixed formulas: one-third of the distance along the base and one-third of the way up the height.
• The coordinates of the centroid give you a specific point, acting as the spot where, if you could support the shape, it would balance perfectly.
• By knowing the centroid, you can find out how far that point travels when the shape is rotated around an axis, which is crucial for applying Pappus' Theorem.

In the given exercise, the centroid helps guide the motion of the triangular shape as it spins around the axis, acting as our benchmark to understand how much space the rotating shape occupies.

Solid of Revolution

A solid of revolution is a three-dimensional object created by rotating a two-dimensional shape around an axis. This concept explains the connection between movement and geometry.

• Imagine taking a flat shape, like a triangle, and spinning it around a straight line. This spinning creates a 3D object, resembling common items like the shape of a vase or a bowl.
• Through the rotation, every point on the shape traces a circular path around the axis, forming a continuous, symmetrical volume.
• These solids are everywhere in the real world, from machine parts to decorative elements, illustrating the importance and applications of science and math in tangible forms.

By understanding how solids of revolution are formed, you can approach such problems with confidence, identifying axis rotation points and leveraging them to solve for volumes in an intuitive way. The exercise demonstrates how something as simple as rotating a line can create a complex and useful 3D object.