652
Chapter 6 | Applications of Integration
6.2 EXERCISES 58. Derive the formula for the volume of a sphere using the slicing method. 59. Use the slicing method to derive the formula for the volume of a cone. 60. Use the slicing method to derive the formula for the volume of a tetrahedron with side length a . 61. Use the disk method to derive the formula for the volume of a trapezoidal cylinder. 62. Explain when you would use the disk method versus the washer method. When are they interchangeable? For the following exercises, draw a typical slice and find the volume using the slicing method for the given volume. 63. A pyramid with height 6 units and square base of side 2 units, as pictured here.
66. A pyramid with height 5 units, and an isosceles triangular base with lengths of 6 units and 8 units, as seen here.
67. A cone of radius r andheight h has a smaller cone of radius r /2 and height h /2 removed from the top, as seen here. The resulting solid is called a frustum .
64. A pyramid with height 4 units and a rectangular base with length 2 units and width 3 units, as pictured here.
For the following exercises, draw an outline of the solid and find the volume using the slicing method. 68. The base is a circle of radius a . The slices perpendicular to the base are squares. 69. The base is a triangle with vertices (0, 0), (1, 0), and (0, 1). Slices perpendicular to the x -axis are semicircles. 70. The base is the region under the parabola y =1− x 2 in the first quadrant. Slices perpendicular to the xy -plane are squares. 71. The base is the region under the parabola y =1− x 2 and above the x -axis. Slices perpendicular to the y -axis are squares.
65. A tetrahedron with a base side of 4 units, as seen here.
This OpenStax book is available for free at http://cnx.org/content/col11964/1.12
Made with FlippingBook - professional solution for displaying marketing and sales documents online