← MATH 01131 · Calculus II
MATH 01131 · Applications
Volumes (disk/washer/shell)
Definition
Disk: V=π∫[R(x)]² dx. Washer: π∫([R_out]²−[R_in]²). Shell: V=2π∫ radius·height·thickness (often 2π∫ x f(x) dx).
Read this
Choose method by washers vs shells convenience relative to the axis of rotation.
Always identify outer/inner radius from the axis — distances, not just function values blindly.
Sketch a slice; label R(x) carefully for rotation about y=k or x=k.
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Try this
Attempt each problem first, then open the worked solution.
1.Volume of y=√x, x=0..4 rotated about x-axis (disks).