1. 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.