Q3 a) to h)
3. Here we consider various solids of revolution whose aris is the diagonal line $y=x$. The goal is to adapt the "pile of thin disks" idea from Question $2(b)$ to this new situation. For each real constant $m \neq 1$, the curve $y=m x+(1-m) x^{2}$ 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 $y=m x+(1-m) x^{2}$. Your sketch should show enongh 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 $\left[m_{0}, m_{1}\right]$ 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 $r(x)$, valid for each number $x$ 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 $\left[m_{0}, m_{1}\right]$ found in part (b). Note: Your formula should reproduce your earlier findings when $m$ is replaced with either 0 or 2. (f) Suppose $m \neq 1$ lies in the interval $\left[m_{0}, m_{1}\right]$ found above. Set up, but do not evaluate, a definite integral whose exact value equals the volume $V(m)$, in terms of the function $r(x)$ found earlier. Hint: The answer is not quite $\int_{0}^{1} \pi r(x)^{2} d x$. (g) Find the exact volume $V\left(m_{0}\right)$, where $m_{0}$ is the smallest number in the interval found in part (b). (h) Find the exact volume $V(0)$.