Problem

一、單選題

6. Which expression represents the volume of the solid generated when the region between the following two curves

$$y=6-x^2 \quad\text{and}\quad y=\frac{x^2}{2}$$

over the interval $[0,2]$ is rotated around the $x$-axis?

(A) $2\pi\int_2^6 y\sqrt{6-y}\,\mathrm{d}y-2\pi\int_0^2 y^{3/2}\sqrt{2}\,\mathrm{d}y$

(B) $2\pi\int_2^6 y\sqrt{6-y}\,\mathrm{d}y+2\pi\int_0^2 y^{3/2}\sqrt{2}\,\mathrm{d}y$

(C) $2\pi\int_0^6 y\sqrt{6-y}\,\mathrm{d}y-2\pi\int_0^2 y^{3/2}\sqrt{2}\,\mathrm{d}y$

(D) $2\pi\int_0^6 y\sqrt{6-y}\,\mathrm{d}y+2\pi\int_0^2 y^{3/2}\sqrt{2}\,\mathrm{d}y$

(E) $2\pi\int_0^6 y\sqrt{6-y}\,\mathrm{d}y-2\pi\int_0^2 y^{1/2}\sqrt{2}\,\mathrm{d}y$。