Cale3 Final Take Home (18F) NAME Due Tuesday (Dec 18) (Show exact value and Show all your work) 1 . Find the volume of the solid E between z 3 x2 + уг) (a cone) and z-f ID x2-y2 (a semi-sphere) by (by any coordinates) evaluating the triple integral Vlldv

The sphere x 2 + y 2 + z 2= 2a 2 and the region inside the cylinder x 2 + y 2 = a 2 Find the volume.

Use​ Pappus's Theorem to find the volume of the solid obtained when the region bounded by y=4x2​, y=​0, and x=​1, whose centroid is 34,310​, is revolved about the​ y-axis. Check your answer by using the method of cylindrical shells. What is the volume of the solid in cubic units?

Use cylindrical coordinates to find the volume of the region bounded by the plane z = 0 and the hyperboloid z= √37-√ 1 + x² + y² Set up the triple integral using cylindrical coordinates that should be used to find the volume of the region as as possible. Use increasing limits of integration. 10*** dz dr de RAMON ⁰77 0

a. Express the volume of the wedge in the first octant that is cut from the cylinder y2+z2=1 by the planes y=x and x=1 as a triple integral. b. Use either the Table of Integrals or a computer algebra system to find the exact value of the triple integral in part (a).