(Solved): HOW and WHY did they get the radius of 12 to equal 72pi? Below, a circular plate is submerged in wat ...

Below, a circular plate is submerged in water. Compute the hydrostatic force against one side of the plate. Use $9.8\mathrm{m}/{\mathrm{s}}^2$ to approximate gravitational acceleration. To compute the hydrostatic force on one side of the plate, we use the following coordinate system. Recall that the density of water is $1000\mathrm{kg}/{\mathrm{m}}^3$ and compute the hydrostatic force. $\begin{array}{rl}F& ={?}_{?12}^{12}1000(9.8)(16?y)\left(2\sqrt{144?y^2}\right)dy\\ & =1000(9.8)(2)\left(16{?}_{?12}^{12}\left(\sqrt{144?y^2}\right)dy?{?}_{?12}^{12}y\left(\sqrt{144?y^2}\right)dy\right)\end{array}$ The second integral is the integral of an odd function over a symmetric interval, so it is equal to 0 , while the first integral is the area of a hemisphere with radius 12 , so is equal to $72?$. Plugging this in, compute $\begin{array}{rl}F& =1000(9.8)(2)(16)(72?)\\ & ?70934648.84\end{array}$