MAT 202 Notes, Fall 2011
American University of Science and Technology · Calculus II · 1 page · Mr. Hussam Nader
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1- Main Formula:
b
V = ∫ A(x) dx
a
This formula is applied in the following 3 methods:
i- Slicing Method:
Take a cross-sectional slice of the shape.
A(x) is found depending on the shape of the cross-section.
Example: Rectangle: A(x) = length * width
Circle: A(x) = π R2
ii- Disk Method:
The cross-section is already known to be a circle.
so A(x) = π R2
b
and V = ∫ π R2 dx
a
iii- Washers Method:
Donut-shaped cross-sections, so the cross-section area is:
Area of Outer Circle - Area of Inner Circle
A(x) = π R outer
2 - π R inner
2
b
and so, V = ∫ π R outer2 - π R inner2 dx
a
b
that is, V = ∫ π ( R outer
2 - R inner
2 ) dx
a
2- Cylindrical Shells Formula:
b
and V = ∫ Circumference * height dx
a
b
and V = ∫ 2 π R * height dx
a