Answer
$288 \pi$
Work Step by Step
The shell method to compute the volume of a region: The volume of a solid obtained by rotating the region under $y=f(x)$ over an interval $[m,n]$ about the y-axis is given by:
$V=2 \pi \int_{m}^{n} (Radius) \times (height \ of \ the \ shell) \ dy=2 \pi \int_{m}^{n} (y) \times f(y) \ dy$
Now, $V=2\pi \int_{1}^{7} x (-x^2+8x-7) \ dx \\= 2\pi \int_{1}^{7} (-x^3+8x^2-7x) \ dx \\= 2 \pi [-\dfrac{x^4}{4}+\dfrac{8x^3}{3}-\dfrac{7x^2}{2}]_{1}^{7} \\=2 \pi (\dfrac{1715}{12}+\dfrac{13}{12}) \\= 288 \pi$
