Mathematics IA Assignment 6 Semester 1, 2016 Submit solutions by 2pm Monday May 9th (together with solutions for Assignment 7) Algebra Consider the following subsets of R2 : C1 = {(x, y) R2 | x 0} C2 = {(x, y) R2 | x + y 1} Draw a sketch of C1 C2 and C1 C2 ; state whether each set is convex or not. If the set is not convex, give an example of a line segment for which the definition of convexity breaks down. Calculus 1. Consider a sphere of radius R. Using a side-on view, we can picture the sphere using the circle of radius R centred at the origin in R2 , described by the equation x2 + y 2 = R2 . (a) Imagine taking a very thin slice of this sphere of thickness dx through the point x on the horizontal axis. So face-on, this slice looks like a circle, and (ignoring the fact that the edge is not quite square), the slice can be regarded as a cylinder of thickness dx. Write down the volume of this very thin slice , expressing it in the form A(x) dx for some function A(x). (b) Now write down an integral that calculus the volume of the entire sphere by \"adding up\" the volumes of all these very thin slices, and use the Fundamental Theorem of Calculus to evaluate that integral. 1