Problem 43 - Thomas Calculus

Question

Derive the formula for the volume of a right circular cone of height \(h\) and radius \(r\) using an appropriate solid of revolution.

Step-by-Step Solution

Answer

The volume of the cone is \( V = \frac{1}{3} \pi r^2 h \).

Step 1: Introduction to Solid of Revolution

To derive the volume of a right circular cone, we can consider it as a solid of revolution. We will rotate a line segment about the x-axis to form the cone.

Step 2: Setting Up the Line Segment

The line segment rotates around the x-axis, represented by the linear equation in the form: \[ y = \frac{r}{h}x \]where \(y\) is the radius at any point \(x\) and \(x\) ranges from 0 to \(h\).

Step 3: Using the Disk Method for Volume

For a small thickness \(dx\) located at a distance \(x\) from the origin, the volume of the disk (slice of the cone) is:\[ dV = \pi y^2 dx \]Substitute the equation for \(y\):\[ dV = \pi \left(\frac{r}{h}x\right)^2 dx \]

Step 4: Integrating to Find Total Volume

To find the total volume \(V\) of the cone, integrate the expression for \(dV\) from 0 to \(h\):\[ V = \int_0^h \pi \left(\frac{r}{h}x\right)^2 dx \]This simplifies to:\[ V = \pi \left(\frac{r^2}{h^2}\right) \int_0^h x^2 \, dx \]

Step 5: Evaluate the Integral

Evaluate the integral\[ \int_0^h x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^h = \frac{h^3}{3} \]Therefore, the volume is:\[ V = \pi \frac{r^2}{h^2} \cdot \frac{h^3}{3} = \frac{1}{3} \pi r^2 h \]

Step 6: Conclusion

We've derived the formula: the volume \(V\) of a right circular cone of height \(h\) and radius \(r\) is:\[ V = \frac{1}{3} \pi r^2 h \]

Key Concepts

Solid of Revolution

The concept of a solid of revolution is key in geometry and calculus. It's about creating a three-dimensional shape by rotating a two-dimensional shape around an axis. Imagine you have a line segment or a curve, and you spin it around a specific line, like spinning a pencil.
This rotational action produces a solid shape. When you apply this idea, you generate various shapes, including cones, cylinders, or even more complex figures.
The solid of revolution provides an intuitive way to visualize and calculate volumes of such objects by utilizing the symmetry induced by the rotation.

• In our case, the cone is formed by rotating a line segment around the x-axis.
• This understanding simplifies complex volume calculations, especially for symmetrical objects.

Disk Method

The disk method is a technique applied in calculus to find the volume of a solid of revolution. Think of slicing the solid perpendicular to the axis of rotation. Each slice is a disk, hence the name.
By calculating the volume of many small disks and adding them together, you approximate the volume of the entire solid. Each disk has a tiny thickness, and its radius is determined by the function that you are rotating.
In the case of the cone, for each small thickness, we calculate:

• The disk's radius at a distance \( x \) from the axis, derived from the function \( y = \frac{r}{h}x \).
• The area of the disk is \( \pi y^2 \) since it's a circle.
• Volume of a thin disk is \( dV = \pi y^2 dx \).

Using this method, we sum the volumes of all disks from 0 to height \( h \) to find the total volume of the cone.

Definite Integral

A definite integral in calculus allows us to calculate the total accumulation of quantities, such as area under a curve or volume. It is represented as the limit of summing many incremental quantities. For the volume of our cone, we integrate the expression for the volume of the disks.

• The integral used is \( V = \int_0^h \pi \left(\frac{r}{h}x\right)^2 dx \).
• The result of this integration gives you an exact value for the cone's volume.

The computation simplifies to a formula through integration techniques. It yields the well-known formula \( V = \frac{1}{3} \pi r^2 h \). This part of calculus provides a powerful tool for deriving exact quantities in complex geometrical contexts.