Find the amount of work needed to pump water out of full spherical tank

(a) 
\[W=\int_{-1}^{1} HdF =\int_{-1}^{1}(2-y )mg \pi x^2 dy=mg \pi \int_{-1}^{1}(2-y)x^2dy\]
\[=mg \pi \int_{-1}^{1}(2-y)(1-y^2)dy =mg \pi \int_{-1}^{1} 2-y-2y^2+y^3dy\]
\[=mg \pi (2y-\frac{y^2}{2}-\frac{2y^3}{3}+\frac{y^4}{4})|_{-1}^{1}=mg \pi [2 (2-\frac{2}{3})]\]
\[=\frac{8}{3}mg \pi =\frac{8}{3} (62.4)\pi=522.76\]
(b)
\[\frac{4}{3}\pi \times mg=\frac{4}{3}\pi \times(62.4)= 261.38\]
(c)
\[W=F\times d=261.38 \times 2 =522.76.\]

As you see we get the same answer in parts (a) and (c).