To find the volume of a solid that is rotated around the y-axis, we can utilize the method of cylindrical shells or the disk/washer method. Here’s a detailed explanation of both methods:
Cylindrical Shells Method
This method is particularly useful when the solid is defined as a function of x. The volume (V) can be calculated using the following integral formula:
V = 2π ∫_a^b x f(x) dx
Where:
f(x) = the height of the shell, which is the function we are rotating
x = the radius of each cylindrical shell
[a, b] = the bounds of integration
Steps:
- Identify the function f(x) that you wish to rotate.
- Determine the limits of integration a and b.
- Set up the integral using the formula mentioned above.
- Evaluate the integral to find the volume.
Disk/Washer Method
This method is apt for cases where the solid is defined as a function of y. For this approach, the formula is:
V = π ∫_c^d [R(y)]² dy
Where:
R(y) = the radius of the disk or washer, which is a function we define in terms of y
[c, d] = the bounds of integration
Steps:
- Identify the function R(y) that describes the shape of the solid.
- Determine the limits of integration c and d.
- Set up the integral using the formula for volume.
- Evaluate the integral to find the volume.
Example
Let’s say we want to find the volume of the region bounded by the curve y = x² rotated about the y-axis between y = 0 and y = 1 using the Disk/Washer Method:
1. Solve for x:
From y = x², we find x = √y.
2. Set up the integral:
V = π ∫_0^1 (√y)² dy = π ∫_0^1 y dy
3. Evaluate:
V = π * [1/2 * y²] from 0 to 1 = π * (1/2 – 0) = π/2.
So, the volume is π/2 cubic units.
In conclusion, whether you choose the cylindrical shells method or the disk/washer method depends on how the solid is defined. Both methods are effective, and understanding each allows you to visualize the process of finding the volume of solids of revolution.