Answer the following vector calculus problems. Be sure to solve all parts. Thank you!

2. Consider a tank in the form of a right circular cylinder having length L and base radius R with a hemisphere at one end. Suppose the tank is partially lled with water such that when tilted the surface of the water cuts through the center of the cylinder like in the gure below. The goal of this problem is to study the volume of the water in the tank. A A (a) Tilted Tank (b) Sideview of Tilted Tank (a) (20 pts) Consider the upper hemisphere of 3:2 + y2 + 22 = R2. The plane 2 = am with a > 0 cuts the hemisphere into two pieces. Use triple integral to find the volume of the smaller piece and use it to deduce the volume of the bigger piece. Hint: Use spherical coordinates. You might need to use the fact that / \\/fu_2 = arcsin(u) + C. n (b) (15 pts) Find the volume of the water inside the tank. Hint: You may want to situate the tank so that the hemispherical cap is above the say-plane just like in part (a) . Then nd a plane that represents the surface of the water (you should probably mentally freeze the water) . (c) (15 pts) Some of the water were drained out so that the surface of the remaining water when tilted cuts through the pole of the hemispherical cap and rim of the cap as in the gure below . How much water were drained? Hint: Worksheets. a Figure 64: Sideview of Partially Drained Tilted Tank