Answer
$1.3219$
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, $V=2\pi \int_{0}^{1.3768} (x)[\sin x^2-\dfrac{x^2}{2}] \ dx\\= \pi \int_{0}^{1.3768} [2x \sin x^2-x^3] \ dx \\= \pi [-\cos x^2 -\dfrac{ x^4}{4}]_{0}^{1.3768} \\= \pi [0.3191-0.8983 +1] \\=1.3219$
