a. around the x-axis

b. around the line y=-2

c. around the line x=ln(1/2)

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y=\sqrt{x-3} \quad y=\sqrt{-x+6} .a) around x-axisshell radius =yShell height=6-y^{2}-\left(y^{2}+3\right)=3-2 y^{2}Volume v=\int_{a}^{b} e \pi (shell-radius) (shell height)dyV=\int_{0}^{1.225} 2 \pi y\left(3-2 y^{2}\right) d yb) around line y=-2.\begin{array}{l}\text { Shell radius }=y-(-2)=y+2 . \\\text { Shell height }=3-2 y^{2} \\V=\int_{0}^{1.225} 2 \pi(y+2)\left(3-2 y^{2}\right) d y\end{array}c)\begin{aligned}\Rightarrow & R(y)=6-y^{2}-\ln (0.5) \\& \gamma(y)=y^{2}+3-\ln \left(\frac{1}{2}\right) . \\V= & \int_{a}^{b} \pi\left[R(y)^{2}-\gamma(y)^{2}\right] d y \\V= & \int_{0}^{1.225} \pi\left[\left(6-y^{2}-\ln (0.8)\right)^{2}-\left(y^{2}+3-\ln \left(\frac{1}{2}\right)\right]^{2} d y\right.\end{aligned} ...