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Hand_out - x h x C or V = âˆ b a dy y h y C â†’ integrate...

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The Volume Problem : Rotate a region about the axis to obtain a solid. Compute the volume of this solid. Disk/Washer Method V = b a dx x A ) ( or V = b a dy y A ) ( integrate ALONG the axis of rotation . Step 1 : Sketch the region. Step 2 : Rotate the region about the given axis to obtain a solid. Step 3 : Slice the solid ORTHOGONALLY to the axis of rotation to get a cross- section. The cross-section can be a disk : area = π r 2 = A(x) or A(y) a washer : area = π (r 1 ) 2 - π (r 2 ) 2 = A(x) or A(y) Cylindrical Shell Method V = b a dx
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Unformatted text preview: x h x C ) ( ) ( or V = âˆ« b a dy y h y C ) ( ) ( â†’ integrate ALONG the axis orthogonal to the axis of rotation . Step 1 : Sketch the region. Divide the region into n rectangles PARALLEL to the axis of rotation. Step 2 : Rotate the region about the given axis to obtain a solid. Step 3 : Each rotated rectangle becomes a cylindrical shell. The cylindrical shell has a base : circumference = 2 Ï€ r = C(x) or C(y) a height : h(x) or h(y) Thin shell r h Thin shell r r 1 r 2...
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