QUESTION

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# Let $f$ be the function defined by $f(x)=\left\{\begin{array}{ll}\frac{1}{2}(x+2)^{2} & \text { for }-2 \leq x<0 \\ 2-2 \sin \sqrt{x} & \text { for } 0 \leq x \leq \frac{x^{2}}{4}\end{array}\right.$. The graph of $f$ is shown in the figure above. Let $\boldsymbol{R}$ be the region bounded by the graph of $\boldsymbol{f}$ and the $\boldsymbol{z}$-axis. (a) Find the area of $\boldsymbol{R}$ (b) Region $\boldsymbol{R}$ is the base of a solid. For this solid, each cross section perpendicular to the $\boldsymbol{m}$. axis is a square. Write, but do not evaluate, an expression involving one or more integrals that gives the volume of the solid. (c) The portion of the region $\boldsymbol{R}$ for $\mathbf{1} \leq \boldsymbol{v} \leq \boldsymbol{2}$ is revolved about the $\boldsymbol{\pi}$-axis to form a solid. Find the volume of the solid. Let f be the function defined by F(x)=(1/2)(x+2)^2 for [-2,0) and 2-2sin(sqrtx) for [0, (x^3)/4]. the graph of f is shown in the figure above. Let R be the regiok bounded by the graph of f and the x-axis.  