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7.3.3 Volume by Cross-sectional Areas A.K.A. - Slicing

7.3.3 Volume by Cross-sectional Areas A.K.A. - Slicing. I. Slicing. It is possible to find the volume of a solid (not necessarily a SOR) by integration techniques if parallel cross-sections obtained by slicing solid with parallel planes perpendicular to an axis have the same basic shape.

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7.3.3 Volume by Cross-sectional Areas A.K.A. - Slicing

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  1. 7.3.3 Volume by Cross-sectional AreasA.K.A. - Slicing

  2. I. Slicing It is possible to find the volume of a solid (not necessarily a SOR) by integration techniques if parallel cross-sections obtained by slicing solid with parallel planes perpendicular to an axis have the same basic shape. If the area of a cross-sectionis known and can be expressed in terms of x or y, then the area of a typical slice can be determined. The volume can be obtained by letting the number of slices increase indefinitely.

  3. Therefore,

  4. II. Examples A.) Assume that the base of a solid is the circle and on each chord of the circle parallel to the y-axis there is erected a square. Find the volume of the resulting solid.

  5. B.) Find the volume of the solid whose base is the region in the first quadrant bounded by , the x-axis, and the y-axis, and whose cross-sections taken perp. to the x-axis are squares.

  6. C.) Find the volume of the solid whose base is between one arc of y = sin x and the x-axis, and whose cross-sections perp. to the x-axis are equilateral triangles.

  7. III. Other Links • http://mathdemos.gcsu.edu/mathdemos/sectionmethod/sectiongallery.html • http://www.ies.co.jp/math/java/calc/index.html • http://www.geocities.com/pkving4math2tor7/7_app_of_the_intgrl/7_03_01_finding_vol_by_slicing.htm • http://www.geocities.com/pkving4math2tor7/7_app_of_the_intgrl/7_03_02_finding_vol_by_using_cylind_shells.htm

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