Explanation:
Let's first consider a rotation about a vertical axis (x=0 or x=a).
Using washer method, the volume is:
∫ᵧ₁ᵞ² π (x₂² − x₁²) dy
∫ᵧ₁ᵞ² π ((x₂ − a)² − (x₁ − a)²) dy
Using shell method, the volume is:
∫ₓ₁ˣ² 2π x (y₂ − y₁) dx
∫ₓ₁ˣ² 2π (x − a) (y₂ − y₁) dx
First, notice that with washer method, the differential dy is parallel to the axis of rotation. With shell method, the differential dx is perpendicular to the axis of rotation.
Also, notice that the limits of integration must match the differential.
Choosing the best method will depend on the function.
If x² can be written in terms of y, and integrated, then washer method is the one to use.
If y can be written in terms of x, and integrated, then shell method is the one to use.
In other words, use washer method if x = f(y) is simple to integrate. Use shell method if y = f(x) is simple to integrate.
If instead we rotate about a horizontal axis (y=0 or y=a), the volume with washer method is:
∫ₓ₁ˣ² π (y₂² − y₁²) dx
∫ₓ₁ˣ² π ((y₂ − a)² − (y₁ − a)²) dx
Using shell method, the volume is:
∫ᵧ₁ᵞ² 2π y (x₂ − x₁) dy
∫ᵧ₁ᵞ² 2π (y − a) (x₂ − x₁) dy
We choose washer method when y² = f(x) is easy to integrate.
We choose shell method when x = f(y) is easy to integrate.
So here's the takeaway:
If the axis of rotation is x = 0 or x = a, and x² = f(y) is easy to integrate, use washer method.
If the axis of rotation is y = 0 or y = a, and y² = f(x) is easy to integrate, use washer method.
Otherwise, use shell method.