1. 7.3.3 Volume by Cross-sectional Areas A.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-section is 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.

7. 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.

10. 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.

13. III. Other Links  http://mathdemos.gcsu.edu/mathdemos/sectionmetho d/sectiongallery.html http://mathdemos.gcsu.edu/mathdemos/sectionmetho d/sectiongallery.html  http://www.ies.co.jp/math/java/calc/index.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_01_finding_vol_by_slicing.htm  http://www.geocities.com/pkving4math2tor7/7_app_ of_the_intgrl/7_03_02_finding_vol_by_using_cylin d_shells.htm http://www.geocities.com/pkving4math2tor7/7_app_ of_the_intgrl/7_03_02_finding_vol_by_using_cylin d_shells.htm