The complete elliptic integral of the first kind is the integral ,where 0 < k < 1 is constant.a. Show that the first four terms of the binomial series for are .b. From part (a) and the reduction integral Formula 67 at the back of the book, show that .

ME5331 Lecture 15 Line Integrals Example 1. Evaluate the following integral along each of the paths shown in the figure. Solution: We have four paths of integration: (a) along the parabolic arc that joins the points A and B; (b) along the straight line AB; (c) along the path APB; (d) along the path AQB. Since we need to integrate with respect to x, the limits of integration for all paths are from A, to B, First we need to express in terms of . To do this, we recall from the definition of a line integral that the integrand is always evaluated along the path of integration. (a) Along the parabolic arc that joins the points A and B, we have . Thus, or