l. N830? Honors Problem 4 Use integral calculus to compute the volume of a spherical cap. which is the region inside the sphere 1:2 + y'2 + 22 = R2 and above the plane 3 = (2. Here assume that I] :1 o 5 R. Draw a picture that illustrates how you got to the integrand and the integration domain. The result should be 2 . . . . g1lili'i aoRz + :cr'i. Vcr] = 3 3 . To derive the formula [or the volume of the sphere above 2: = a: when (2 s: I] subtract the volume of the cap below 2: = o from the volume of the sphere. You should obtain that the above formula also holds when R 5 or 5 I]. . In physics work is the integral of force over distance. 'We want to compute the work to lift a stone sphere out of water. Initially. the sphere is totally submerged in water with the north pole touching the water surface. At the end._ the sphere is in the air with the south pole touching the water. In between. the gravitational force is F = Wang + nglm pm] where g gravitational constant. V" volume of the part of the sphere in the air. if\". volume of the part of the sphere in the water. p, densit'l,r of mass for the sphere. In\". densit'l,r of mass for water. The lirst term represents the gravitational force of the sphere above water and the second term the force of the part below water. In the latter the displaced water is subtracted. Express the force as a function of z=position of the north pole. where z = [l is the water line. Find Va and V5 as functions of 2. Then compute the work as an integral over the zvariable