Calculus I · Unit 3B · lesson
Disks and Washers
Learn Disks and Washers through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Section overview
Area and volumeWhat this section is building
Learn Disks and Washers through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Area adds thin rectangles, slicing adds cross-sectional slabs, washers add annular slabs, and shells add thin cylindrical walls.
Sketch the region and axis, test vertical and horizontal slices, and choose the description that stays single-valued with the fewest interval splits.
Measuring a radius from the wrong curve, subtracting boundaries in the wrong order, or mixing a shell radius with its height.
Learning objectives
Construct radius functions and integrate cross-sectional areas for solids of revolution.
Disks and Washers
Radii are distances from the axis of rotation
Disks and washers arise from slices perpendicular to the axis of rotation. The radius is not automatically the function value; it is the distance from the axis to the boundary. If the region does not touch the axis, rotation leaves a hole, and the cross-sectional area is outer disk minus inner disk: .
A labeled representative slice prevents most mistakes. Mark the axis, the outer radius, the inner radius, and the thickness before integrating. If the axis is shifted, use distance expressions such as rather than copying . Square the radii only after those distances are correct. The geometry decides the formula; the algebra merely carries it out.
Rotating a region around an axis creates circular cross-sections. A disk has area . A washer has area
Rotate a region around the x-axis
Rotate the region under from to around the -axis.
Worked solution
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u3b-disk-01Find the volume obtained by rotating , , around the x-axis.
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Use disks with radius .
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Washer geometry. A vertical strip rotated around the x-axis forms a washer with outer and inner radii. Show outer radius, inner radius, and thickness for rotation around a horizontal axis. Measure radii from the axis, not from the other curve.
Every relationship in washer geometry uses written labels together with distinct line styles, markers, or fill patterns; color is never the only carrier of meaning.
Why it matters: Show outer radius, inner radius, and thickness for rotation around a horizontal axis.
Washer geometry. Show outer radius, inner radius, and thickness for rotation around a horizontal axis.
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