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

Short answer

Based on the given step-by-step solution, find the volume of the solid that is left after drilling a hole with radius 2 along the axis of a right circular cylinder of height 6 and radius 4. Answer: 48π cubic units

Step-by-step solution

Identify the limits of integration for the smaller and the larger cylinders

Since the hole (smaller cylinder) has a radius of 2, its limits of integration will range from 0 to 2. The larger cylinder has a radius of 4; its limits of integration will range from 0 to 4.

Set up the shell volume formula

The shell method formula for volume is: V = 2π ∫[R(x)h(x)]dx, where R(x) is the radius of a cylindrical shell at position x, h(x) is the height of the shell, and the integral is taken from the limits identified in step 1.

Calculate the volume of the hole (smaller cylinder)

The radius function of the smaller cylinder, R(x), is 2 because the hole has a radius of 2. The height function for both cylinders is constant and equal to 6. Therefore, the volume of the hole is: V_hole = 2π ∫[R(x)h(x)]dx = 2π ∫[2*6]dx from 0 to 2 V_hole = 2π(12) ∫dx from 0 to 2 V_hole = 24π (2 - 0) = 48π cubic units.

Calculate the volume of the larger cylinder

The radius function of the larger cylinder, R(x), is x as the radius increases along the x-axis, and the height function h(x) is 6 since it is constant. Thus, the volume of the larger cylinder is: V_cylinder = 2π ∫[R(x)h(x)]dx = 2π ∫[x*6]dx from 0 to 4 V_cylinder = 12π ∫x dx from 0 to 4 V_cylinder = 12π (1/2)(4^2 - 0^2) = 96π cubic units.

Subtract the volume of the hole from the volume of the larger cylinder

Finally, we can find the volume of the solid formed by subtracting the volume of the hole from the volume of the larger cylinder: V_solid = V_cylinder - V_hole = 96π - 48π = 48π cubic units. So the volume of the solid formed by drilling a hole with radius 2 along the axis of a right circular cylinder of height 6 and radius 4 is 48π cubic units.

Key concepts

Volume of Solids

When trying to find the volume of a solid, especially a complex one, it's important to break it down into parts that are easier to handle. Here, we use the concept of a solid that has been modified by removing a portion, such as creating a hole within it. This method allows us to calculate the volume by subtracting the volume of the removed part (like a drilled hole) from the total volume of the original solid. By focusing on these individual volumes, we can utilize various techniques like integration to precisely determine the total volume. This method ensures accuracy even when dealing with intricate shapes and modifications.
Understanding these concepts is crucial when dealing with problems that involve the creation or removal of parts. In our example, the original solid is a cylinder, and the part removed is another cylindrical shape, leading us to a unique situation that is best handled with the shell method.

Cylindrical Shells

The concept of cylindrical shells is both fascinating and practical, especially when dealing with rotational bodies. Imagine stacking a series of thin cylindrical "shells" one inside the other to create a full 3D solid. This approach allows you to find the volume of a complex shape by analyzing these simpler components. In our problem, the shells represent the thickness of the cylinder at various radii.
Using the shell method involves integrating these shells over a defined region. Each shell has a "radius," which changes depending on its position in the larger cylinder, and a "height," which remains constant as described in the problem.

• The formula for the volume of a cylindrical shell is: \( V = 2\pi \int R(x)h(x) \, dx \).
• Here, \( R(x) \) represents the radius at point \( x \), and \( h(x) \) is the height.

This method is particularly useful for objects that have symmetrical cross-sections or holes, allowing for accurate volume calculations by integrating along the axis.

Integration Limits

Setting the proper integration limits is essential for accurate calculations when applying the shell method. Your integration should cover the exact region you're interested in, which means understanding the geometry of the solid. In this exercise, there are two important cylinders: the smaller one (hole) and the larger one (original solid). Each has its own integration limits, which define the start and end points of your integration process along the x-axis.

• The smaller cylinder, which forms the hole, has limits from 0 to 2.
• The larger cylinder ranges from 0 to 4.

By carefully determining these boundaries, you ensure that the integral only calculates the shell parts that are actually contributing to the volume calculation. These limits reflect the geometry's bounds and help isolate the correct region in the shell integration process.

Cylinder Geometry

Understanding cylinder geometry is key when working with these calculations. A cylinder is characterized by its radius and height. Here, we have a solid cylinder with a larger outer radius of 4 and a height of 6. Inside, a smaller cylinder (the hole), centered along the same axis, has a radius of 2.
Each cylinder can be described by:

• Its radius, which dictates the extent from the center to any point along the edge.
• Its height, providing the vertical length of the cylinder.

For the shell method, the geometry dictates how we set up our integration. The larger cylinder has a variable radius across the shell, while the hole retains a constant radius of 2 across its section. This setup allows us to form precise integrals to determine each part's volume, considering both the simple structure of cylinders and their geometrical properties.