Problem 28 - Calculus Early Transcendentals

Question

Use the shell method to find the volume of the following solids. The solid formed when a hole of radius 2 is drilled symmetrically along the axis of a right circular cylinder of height 6 and radius 4.

Step-by-Step Solution

Answer

Answer: The volume of the solid is 72π cubic units.

Step 1: Finding the volume of the outer cylinder

In order to find the volume of the outer cylinder, we can use the following formula: Volume = π × (radius_outer)^2 × height Here, the outer radius (radius_outer) is 4 and the height is 6. Volume_outer = π × (4)^2 × 6

Step 2: Finding the volume of the inner cylinder

To find the volume of the inner drilled cylinder, we can use the same volume formula as in Step 1. Here, the inner radius_of_inner (radius_inner) is 2 and the height is also 6. Volume_inner = π × (2)^2 × 6

Step 3: Finding the volume of the solid

To find the volume of the solid, we need to subtract the volume of the inner cylinder from the volume of the outer cylinder. Volume_solid = Volume_outer - Volume_inner Plug in the values from Step 1 and Step 2: Volume_solid = (π × (4)^2 × 6) - (π × (2)^2 × 6)

Step 4: Simplifying the expression

Now we can simplify the expression and find the volume of the solid. Volume_solid = (16π × 6) - (4π × 6) Volume_solid = 6π(16 - 4) Volume_solid = 6π × 12

Step 5: Finalizing the volume of the solid

Multiplying the numbers, we get the final volume of the solid: Volume_solid = 72π cubic units So, the volume of the solid formed when a hole of radius 2 is drilled symmetrically along the axis of a right circular cylinder of height 6 and radius 4 is 72π cubic units.

Key Concepts

Volume of Solids

Understanding how to find the volume of solids is a crucial concept in geometry, especially when dealing with shapes like cylinders and other 3D objects. The volume of a solid is the amount of space it occupies.
For complex shapes, breaking them down into simpler parts such as basic cylinders or boxes often makes calculations easier.

• For regular shapes, standard formulas are used to calculate volume.
• When dealing with irregular solids, integration techniques like the shell method can be applied.
• The volume is usually expressed in cubic units, like cubic centimeters or cubic meters.

The shell method is particularly useful in calculus for finding the volume of solids of revolution, where a shape is rotated around an axis.

Cylinder Volume

A cylinder is a simple geometric shape that holds a lot of importance when calculating volume. The basic formula for the volume of a cylinder involves its radius and height.
The formula is given by:
\[\text{Volume} = \pi \times (\text{radius})^2 \times \text{height}\]This formula works because:

• \(\pi \cdot (\text{radius})^2\) calculates the area of the circle at the base.
• The height extends this area into the third dimension to calculate volume.

So, in the case of our problem, both the outer and inner cylinders are calculated using this simple yet powerful formula. Understanding this calculation is essential for solving problems involving cylinders.

Drilled Hole in Cylinder

When a hole is drilled through a cylinder, like in this example, calculating the remaining volume requires subtracting the volume of the hole from the volume of the original cylinder.
This problem starts by setting up the volumes for both:

• The outer cylinder: This represents the entire shape before the hole is drilled.
• The inner cylinder (the hole): Subtracting this from the outer cylinder gives the volume of the solid with the hole.

This subtraction is straightforward:\[\text{Volume of solid} = \text{Volume of outer cylinder} - \text{Volume of inner cylinder}\]Understanding this approach helps in visualizing and calculating more complex real-life problems where similar operations might be necessary, providing a solid grasp of three-dimensional space manipulation.