Answer
$W$ = $9800l\pi{r}^{3}$ $J$
Work Step by Step
at location $y$
length = $l$
width = $2\sqrt {r^{2}-y^{2}}$
thickness = $Δy$
$V$ = $(l)(2\sqrt {r^{2}-y^{2}})(Δy)$
$F$ = $(9.8)(1000)(2l\sqrt {r^{2}-y^{2}}Δy)$
$F$ = $19600l\sqrt {r^{2}-y^{2}}Δy$
distance = $r-y$
$V$ = $\int_{-r}^{r}19600l\sqrt {r^{2}-y^{2}}(r-y)dy$
$V$ = $\int_{-r}^{r}19600lr\sqrt {r^{2}-y^{2}}dy$ - $\int_{-r}^{r}19600ly\sqrt {r^{2}-y^{2}}dy$
$\int_{-r}^{r}19600lr\sqrt {r^{2}-y^{2}}dy$ = $\frac{1}{2}\pi{r}^{2}$
$\int_{-r}^{r}19600ly\sqrt {r^{2}-y^{2}}dy$ = $0$
$V$ = $\frac{1}{2}\pi{r}^{2}$
total work
$W$ = $19600lr(\frac{1}{2}\pi{r}^{2})$-$19600l(0)$
$W$ = $9800l\pi{r}^{3}$ $J$
