Twelve people join hands for a circle dance. In how many ways can they do this

Chapter 6: Applications of Definite Integrals ____________________________________________________________ 6.1 Volumes Using Cross-Sections 1.Method of Slicing -- The cross-section is perpendicular to the x-axis. -- If Δx is small enough, each slice is close to a right cylinder -- The whole volume is n S ≈ ∑ A(x kΔx k = 1 -- where n is the number of slices n S = ln→∞k = 1(xk)Δx b S = ∫ A(x) dx a -- where A is the area of the cross-section at x Steps to find the volume of a solid 1. Sketch a graph of the solid (optional but highly recommended) 2. Sketch the cross-section and find the formula for its area 3. Find the limits 4. Compute the integral Example 1: Find the volume of a pyramid. - The base is a 10 x 10 in. square located at x = 20 and is perpendicular to the x-axis - The cross-section at x is a square - The length is x/2 Therefore, the area formula is: A(x) =/(x 2= x / Now compute the integral 20 2 S = ∫ x / dx 0 = 1 / · x / ∣ 20 0 = 2000/3 Example 2: A solid lies between planes perpendicular to the x-axis at x = -1 and x = 1. The cross-section perpendicular to the x-axis are equilateral triangles whose bases run from y = − √ 1− x2 to 2 y = √ 1 −x . Find the volume. Example 2: A solid lies between planes perpendicular to the x-axis at x = -1 and x = 1. The cross-section perpendicular to the x-axis are 2