I'd like to ask what have I been doing wrong.
Determine the volume generated by revolving the region bounded by the curves defined by $ y=1+x^2$ and $y=5$ about the $x$-axis.
I got the limits of integration to be $$x=2$$ $$x=-2$$
My solution is
$$ V=\pi \int_{-2}^{2} 5^2-(2+x^2)^2$$
I arrived at $$(2+x^2)^2$$ because I added 1 since the vertex of the curve is at (0,1) and it needs to revolve about the x-axis
Is that wrong?
There is no reason for you to add that $1$. The answer is$$\pi\int_{-2}^25^2-(x^2+1)^2\,\mathrm dx=\frac{1\,088\pi}{15}.$$