Problem 29 - Calculus Early Transcendentals

Question

Use the shell method to find the volume of the following solids. The solid formed when a hole of radius 3 is drilled symmetrically along the axis of a right circular cone of radius 6 and height 9.

Step-by-Step Solution

Answer

Answer: The volume of the remaining solid is \(9\pi\) cubic units.

Step 1: Set up the integral for the volume of the cone

We have a right circular cone with radius 6 and height 9. The equation relating the height and radius of the cone can be found using similar triangles: \[h(x) = \frac{9x}{6}\] The volume of the cone can be found using the shell method, which requires integrating the product of the shell thickness, shell height, and shell radius: \[V_{cone} = \int_0^6 2\pi xh(x)dx = \int_0^6 2\pi x \cdot \frac{9x}{6}dx\]

Step 2: Calculate the integral of the cone volume

Now, we will compute the integral for the cone volume: \[V_{cone} = \int_0^6 2\pi x \cdot \frac{9x}{6}dx = \frac{3\pi}{2}\int_0^6 x^2dx = \frac{3\pi}{2}[\frac{x^3}{3}]_0^6 = 36\pi\]

Step 3: Set up the integral of the cylindrical hole

The hole in the cone has a radius of 3 and is drilled along the axis of the cone, so its height is also 9. Using the shell method again, we can find the volume of the hole: \[V_{hole} =\int_0^3 2\pi xr(x)dx =\int_0^3 2\pi x\cdot \frac{9}{3}dx\]

Step 4: Calculate the integral of the hole volume

Now, we will evaluate the integral for the hole volume: \[V_{hole} = \int_0^3 2\pi x \cdot \frac{9}{3}dx = 6\pi \int_0^3 xdx = 6\pi[\frac{x^2}{2}]_0^3 = 27\pi\]

Step 5: Calculate the volume of the remaining solid

Finally, we will find the volume of the solid by subtracting the hole volume from the cone volume: \[V_{solid} = V_{cone} - V_{hole} = 36\pi - 27\pi = 9\pi\] The volume of the solid with the drilled hole is \(9\pi\) cubic units.

Key Concepts

Shell Method

The Shell Method is a technique in calculus used for finding the volume of a solid of revolution. It involves the integration of cylindrical shells. Each shell is created by rotating a line element around an axis. This method is particularly useful when you have a shape that extends around an axis but is not naturally symmetrical, like the drilled cone in the exercise.

To apply the Shell Method:

• Identify the function or shape you're rotating.
• Determine the axis of rotation.
• Calculate the radius of the shell, which is the distance from the axis of rotation.
• Find the height of the shell, which depends on the function or region being rotated.
• Integrate the product of the circumference of each shell, its height, and its thickness across the specified interval.

This approach allows you to build an integral that represents the entire volume of the complex shape.

Integral Calculus

Integral Calculus is an essential branch of calculus focused on the concept of integrals. It primarily deals with the total size or value, such as areas under a curve, accumulated quantities, or, as in the given problem, volumes of solids. It is the process of finding the integral of a function, which can be thought of as the reverse operation of differentiation.

The definite integral is what we used here to determine volumes. It is defined as a limit of a sum of areas of rectangles (or other shapes) under a curve. The formal expression used in these calculations is:

• The integral sign \( \int \) which denotes integration.
• The function to be integrated, often called the integrand.
• The differential \( dx \), which indicates the variable of integration.
• The interval of integration, set by the lower and upper bounds.

In the example exercise, integral calculus becomes crucial. We used it to calculate both the volume of the entire cone and the volume of the cylindrical hole adjusted to the cone's shape.

Solid of Revolution

A Solid of Revolution is a three-dimensional object created by rotating a two-dimensional shape around an axis. This concept often requires calculating the volume of the object, which is precisely what we do in the exercise.

When tackling problems involving solids of revolution:

• Identify the axis of rotation, as it dictates how the shape will revolve.
• Understand the shape being rotated; this could be a function or even a set of lines combined into a region.
• Choose a method, like the Shell or Disk method, to calculate the volume based on the rotation.
• Set up your integral accordingly, ensuring that it represents the dimensions and shape accurately.

The problem we addressed involved a right circular cone with a drilled cylindrical hole. By visualizing the solid of revolution, we employed the Shell Method to systematically compute the complex volume accurately.