Problem 81 - Thomas Calculus

Question

If lines are drawn parallel to the coordinate axes through a point \(P\) on the parabola \(y^{2}=k x, k>0\) , the parabola partitions the rectangular region bounded by these lines and the coordinate axes into two smaller regions, \(A\) and \(B\) . a. If the two smaller regions are revolved about the \(y\) -axis, show that they generate solids whose volumes have the ratio \(4 : 1 .\) b. What is the ratio of the volumes generated by revolving the regions about the \(x\) -axis? (GRAPH NOT COPY)

Step-by-Step Solution

Answer

The volume ratios are 4:1 for about the y-axis, and equal for about the x-axis.

Step 1: Understand the Problem

The problem requires us to analyze a parabola and its partitions when revolved around axes. A point \( P \) on the parabola \( y^2 = kx \) means when lines are drawn from this point parallel to the axes, they create a rectangle with the axes. We must find the volume of solids formed by revolving the regions \( A \) and \( B \) about the \( y \)-axis and the \( x \)-axis.

Step 2: Set Up Variables

Let \( P(a, b) \) be a point on the parabola \( y^2 = kx \), so \( b^2 = ka \). The rectangle is defined by lines \( x = a \) and \( y = b \). Regions \( A \) and \( B \) are the parts of the rectangle split by the parabola.

Step 3: Calculate the Volume Revolved about the y-axis for Region A

Region \( A \) is below the parabola. By using the disk method, the volume \( V_A \) of the solid formed by revolving region \( A \) is given by the integral \( V_A = \pi \int_0^b (x_{top}^2 - x_{bottom}^2) \, dy \), where \( x_{top} = a \) and \( x_{bottom} = \frac{y^2}{k} \).

Step 4: Calculate the Integral for Region A

The integral becomes \( V_A = \pi \int_0^b \left(a^2 - \left(\frac{y^2}{k}\right)^2\right) \, dy \). Solve this integral to find \( V_A \).

Step 5: Calculate the Volume Revolved about the y-axis for Region B

Region \( B \) is above the parabola. The volume \( V_B \) is \( V_B = \pi \int_b^b y \, a^2 \, dy \). Evaluate this integral.

Step 6: Determine the Ratio for y-axis

Find the ratio \( \frac{V_B}{V_A} \). Simplify to show that \( V_B : V_A = 4:1 \).

Step 7: Calculate the Volume Revolved about the x-axis for Region A

Region \( A \) revolved around the \( x \)-axis uses the cylindrical shell method. The volume is \( V_A' = 2\pi \int_0^b y(a - \frac{y^2}{k}) \, dy \).

Step 8: Calculate the Volume Revolved about the x-axis for Region B

Region \( B \) uses \( V_B' = 2\pi \int_b^{\sqrt{ka}} y(a - \frac{y^2}{k}) \, dy \).

Step 9: Determine the Ratio for x-axis

Find the ratio \( \frac{V_B'}{V_A'} \) and simplify to obtain the ratio of the volumes when revolved around the \( x \)-axis.

Key Concepts

Revolution Volume

When dealing with problems involving revolutions, it's important to understand that these problems often explore how shapes can turn and form a 3D object. In this context, the volume of revolution refers to the space that a 2D shape occupies once it has been turned or revolved around an axis. The common axes of revolution in mathematical problems are the x-axis and y-axis.

These kinds of problems allow us to understand how dimensions switch from two-dimensional space to three-dimensional space. The resulting shape resembles objects with rotational symmetry—such as vases or cups—depending on the initial form and the axis it rotates around. Calculating these volumes requires integral calculus, as we accumulate many small disks or shells that together approximate the solid.

By comprehending the revolution volume, students can grasp how abstract mathematical principles directly apply to real-world objects. It's a fundamental concept not only in pure mathematics but also in fields like engineering and physics.

Disk Method

The disk method is a powerful technique for finding the volume of a solid of revolution. It's used when a solid can be visualized as being composed of stacked circular disks. Imagine a pile of coins as a model; as these disks stack along the axis of revolution, they form a 3D shape.

This method requires revolving a region around a line usually along one of the coordinate axes. For example, if we revolve a region around the x-axis, we calculate the volume with the formula: \[V = \pi \int_a^b [r(y)]^2 \, dy,\]where \(r(y)\) is the function representing the radius of each disk at a specific \(y\) value. Similarly, revolving around the y-axis would involve \(r(x)\).

Key points about the disk method include:

• The integral sums up all the small differences to form the desired volume.
• This method is particularly useful when the cross-section perpendicular to the axis of revolution is a circle without holes.
• It is critical to correctly determine the bounds of integration and understand the axis of revolution effect on the radius function.

Cylindrical Shell Method

The cylindrical shell method offers an alternative to the disk method for finding volumes of solids of revolution. This method visualizes the solid as being made of multiple concentric shells (like nesting Russian dolls), oriented parallel to the axis that the shape revolves around.

When using this method, you revolve a region around a line that might not just be one of the coordinate axes. The key formula is:\[V = 2\pi \int_a^b \, [radius(y) \, height(y)] \, dy,\]where \(radius(y)\) is the distance from the axis of rotation to the shell, and \(height(y)\) is the height of the shell.

This method often simplifies problems where the disk method becomes cumbersome, particularly when the region is parallel to the axis rather than perpendicular:

• The cylindrical shell method is adept at dealing with voids and hollow spaces.
• It is particularly valuable when the radius defined by the distance to the axis of rotation varies.
• Always take care in defining the limits of integration and the specific transformations needed to generate the shells.