Answer
$\dfrac{a^2(a+3) \pi}{3}$
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:
$V=2 \pi \int_{m}^{n} (Radius) \times (height \ of \ the \ shell) \ dy=2 \pi \int_{m}^{n} (x) \times f(x) \ dx$
Now, $V=2\pi \int_{0}^{a} (x+1) (a-x) \ dx\\= 2\pi \int_{0}^{a} (ax-x^2+a-x) \ dx \\=2 \pi [\dfrac{(a-1)x^2}{2} -\dfrac{x^3}{3}+ax]_{0}^{a} \\= 2\pi [\dfrac{a^3-a^2}{2}-\dfrac{a^3}{3}+a^2] \\=\dfrac{a^2(a+3) \pi}{3}$
