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Volume of Revolution

Creating 3D solids by rotating 2D shapes.

We know how to use integrals to find the area under a 2D curve. But calculus allows us to go one dimension higher. By taking a flat shape and rotating it around an axis (like a spinning coin), we can create a 3D object called a Solid of Revolution.

This technique is used to calculate the volume of objects like vases, funnels, pistons, and domes.

[Image of 2D shape rotating to form 3D solid]

1. The Disk Method

Imagine cutting a cucumber into thin slices. Each slice is a circle (a disk). If you find the volume of each disk and add them up, you get the volume of the whole cucumber.

When rotating a curve y = f(x) around the x-axis, each "slice" is a circle with a radius equal to the height of the function, r = f(x).

[Image of disk method slicing diagram]

The area of a circle is πr². Since r = f(x), the area of one slice is π[f(x)]². We integrate this to sum up all the slices.

Formula (Disk Method):
V = π ∫ₐᵇ [f(x)]² dx

Example Problem

Find the volume created by rotating y = √x from x = 0 to x = 4 around the x-axis.

  1. Radius r = √x.
  2. Area of slice = π(√x)² = πx.
  3. Integrate from 0 to 4.
V = π ∫₀⁴ x dx
Antiderivative of x is x²/2.
V = π [ (4²)/2 - (0²)/2 ]
V = π [ 16/2 ] = 8π cubic units.

2. The Washer Method

What if the shape isn't solid? What if it has a hole in the middle, like a donut or a pipe? In this case, our slices aren't solid disks; they are washers (rings).

To find the area of a ring, we take the area of the big circle (Outer Radius, R) and subtract the area of the small hole (Inner Radius, r).

[Image of washer method diagram]
Formula (Washer Method):
V = π ∫ₐᵇ ( [R(x)]² - [r(x)]² ) dx

Note: It is (R² - r²), NOT (R - r)². You must square the radii individually before subtracting.

3. Rotating around Different Axes

We don't always rotate around the x-axis.

  • Rotation around y-axis: Everything flips. The integral is with respect to dy, and equations must be written as x = f(y).
  • Rotation around other lines (e.g., y = -2): The radius changes. You must calculate the distance from the function to the axis of rotation.

4. Real World Application

Engineers use volumes of revolution to design things like turbine blades, medical tubing, and aerodynamic nose cones for aircraft. By defining a precise curve and rotating it, they can calculate the exact material needed for manufacturing.