Volumes of Revolution – Washer Method
Steps for volumes of revolution between two curves (Washer Method):
1.
2.
3.
4.
Identify points of intersection if they are not given. These will be your limits of integration
Graph (Sketch) both functions
Identify which function is greater on the interval.
𝑏
If f(x) > g(x), The volume of the solid formed is ∫𝑎 𝜋[𝑓(𝑥)]2 − [𝑔(𝑥)]2 . Substitute the
functions and limits of integration into this formula
5. Anti-differentiate and solve
1. Find the volume of the solid of revolution generated by revolving the region bounded by f(x)= x2
and g(x) = 4 about the x–axis.
Step 1: Find points of intersection
Step 2: Sketch curves
Step 3: Which function is greater on the
interval?
Step 4: Substitute the functions into the
𝑏
formula: 𝜋 ∫𝑎 [𝑓(𝑥)]2 − [𝑔(𝑥)]2
Step 5: Anti-Differentiate and solve.
Volumes of Revolution – Washer Method
2. Find the volume of the solid of revolution generated by revolving the region bounded by y = x2
and y = 4x – x2 about the x–axis
3. Find the volume of the solid of revolution generated by revolving the region bounded by y = 6 –
2x – x2 and y = x + 6 about the x–axis
4. Find the volume of the solid of revolution generated by revolving the region bounded by f(x) =
sin x, g(x) = cos x, 0 < x < /4