Use the general slicing method to find the volume of the following solid. The solid whose base is the triangle with vertices \( (0,0),(11,0) \), and \( (0,11) \) and whose cross sections perpendicular to the base and parallel to the \( y \)-axis are semicircles Set up the integral that gives the volume of the solid. Use increasing limits of integration. Select the correct choice below and fill in the answer boxes to complete your choice. (Type exact answers,)

To find the volume of the solid whose base is a triangle and cross sections perpendicular to the base are semicircles, we first identify the equation of the line forming the hypotenuse of the triangle, which connects the points \((0,11)\) and \((11,0)\). The slope of this line is \(-1\), leading to the equation \(y = 11 - x\). For a given \(y\), we can find \(x\) using \(x = 11 - y\). The diameter of the semicircle at height \(y\) is \(d = 11 - y\). The radius \(r\) is half of that diameter, so \(r = \frac{11 - y}{2}\). The area \(A\) of a semicircle is given by: \[ A = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi \left(\frac{11 - y}{2}\right)^2 = \frac{\pi (11 - y)^2}{8} \] To find the volume, we integrate this area from the bottom of the triangle \(y = 0\) to the top \(y = 11\): \[ V = \int_{0}^{11} A \, dy = \int_{0}^{11} \frac{\pi (11 - y)^2}{8} \, dy \] So, the integral that gives the volume of the solid is: \[ V = \frac{\pi}{8} \int_{0}^{11} (11 - y)^2 \, dy \]