How to Find the Volume of a Solid Rotated Around the Y Axis

To find the volume of a solid that is rotated around the y-axis, you can use the method of cylindrical shells or the disk/washer method, depending on the function and the bounds you are working with.

Using the Disk/Washer Method

If you are dealing with a function y = f(x) that is rotated around the y-axis, you can use the formula:

V = π ∫ (f(x))² dx

Here, the integral is evaluated from a to b, which are the bounds along the x-axis of the region you are rotating. The term (f(x))² represents the area of the circular cross-section of the solid formed at a given value of x.

Using the Shell Method

Alternatively, if you prefer using the shell method, the volume can be calculated using:

V = 2π ∫ x f(x) dx

In this formula, x is the radius of the shell and f(x) is the height of the shell. Again, you will evaluate the integral from a to b.

Step-by-Step Example

Suppose we want to find the volume of the solid formed by rotating the function y = x², from x = 0 to x = 1.

1. Disk/Washer Method:
• Set up the integral: V = π ∫ (f(x))² dx = π ∫ (x²)² dx = π ∫ x⁴ dx
• Evaluate from 0 to 1: V = π [ (1⁵/5) – (0⁵/5) ] = π/5
2. Shell Method:
• Set up the integral: V = 2π ∫ x (x²) dx = 2π ∫ x³ dx
• Evaluate from 0 to 1: V = 2π [ (1⁴/4) – (0⁴/4) ] = 2π/4 = π/2

Both methods could lead to different results if not used correctly, so choose the method that suits your problem best and ensure you set your limits and functions right. This approach allows you to accurately calculate the volume of solids or objects being formed through rotation.