Describes how to find the volume using the washer technique.

1. Methods of Washers:
• Let f and g be continues and nonnegative on
an interval [a, b]. Let R be the region that is
bounded above y = f(x), below y = g(x), and
the sides by x = a and x = b. The Volume of
the object revolved around the x-axis is 𝑉 =
𝑎
𝑏
𝜋 𝑓 𝑥
2
− 𝑔 𝑥
2
𝑑𝑥
• In the washer method you simply need to
identify the top and bottom curve to evaluate.

2. • Find the volume of the solid of revolution by rotating the region
bounded by the graphs 𝑦 = 𝑥 − 1 and 𝑦 = 𝑥 − 1 2
about the x-
axis
• Notice that the square root function is on top and the parabola on
bottom.
• Notice that the functions intersect at x = 1 and x = 2
• We can know write the following volume formula:
• 𝑉 = 1
2
𝜋 ∙ [ 𝑥 − 1
2
− 𝑥 − 1 2 2
𝑑𝑥

3. Here is how the problem works out
• 𝑉 = 1
2
𝜋 ∙ [𝑥 − 1 − 𝑥 − 1 4]𝑑𝑥 = 𝜋 0
1
[𝑥 − 1 − 𝑥 − 1 4] 𝑑𝑥
• 𝑉 = 𝜋
𝑥2
2
− 𝑥 −
𝑥−1 5
5
2
1
• 𝑉 = 𝜋
22
2
− 2 −
2−1 5
5
−
12
2
− 1 −
1−1 5
5
• 𝑉 = 𝜋 2 − 2 −
1
5
−
1
2
− 1
• 𝑉 = 𝜋
3
10
=
3𝜋
10