Here we consider various solids of revolution whose axis is the diagonal line y = x. The goal is to adapt the "pile of thin disks" For each real constant m 1, the curve y = mx + (1 -m)r is a parabola that crosses the line y = x at the points (0,0) and (1,1). Let R(m) denote the finite region between the parabola and the line. Then let S(m) denote the solid generated by rotating R(m) around the line y = x. We are interested in the volume of the solid S(m): call this volume V(m). (a) Using the same set of axes, draw the line y = x and several of the parabolas y = mx + (1 - m)x. Your sketch should show enough parabolas to communicate all the different types of possible shapes. (b) Suppose m = -6. With reference to a suitable sketch, explain why the "pile of thin disks" idea cannot be used directly to find the volume V(-6). Then determine the largest closed interval [mo, m] of m-values for which this idea can be used directly. (c) Suppose m = 0. Let r(x) denote the radius of the disk of revolution whose centre has coordinates (x,x). Find a formula for r(r), valid for each number z in [0, 1]. Check that the values for r(0) and r(1) are compatible with the geometry of the situation. (d) Repeat part (c), but use m = 2. (e) Extend your reasoning in parts (c)-(d) to produce a formula for r(x) that involves m, and holds for each m (except m = 1) in the interval [mo, m] found in part (b). Note: Your formula should reproduce your earlier findings when m is replaced with either 0 or 2. (f) Suppose m / 1 lies in the interval [mo, m] found above. Set up, but do not evaluate, a definite integral whose exact value equals the volume V(m), in terms of the function r(r) found earlier. r(x) dx. of Hint: The answer is not quite (g) Find the exact volume V(mo), where mo is the smallest number in the interval found in part (b). (h) Find the exact volume V(0).