1. Express the volume of the sphere centered at (0 , 0 , 1) with radius 1 as a t

Question

1. Express the volume of the sphere centered at (0, 0, 1) with radius 1 as a triple integral using a type 3 region (in R3) and then a type II region (in R2).

[Here by type 3 I mean u1(x,z) %u2264 y %u2264 u2(x,z) and (x,z) %u2208 D. Also type II here means g1(x) %u2264 z %u2264 g2(x) and a %u2264 x %u2264 b.] (Do NOT take the integral!)

2. (a) Express the area of the Part of the plane enclosed by the lines y = x, y = %u2212x, and y = 1 as a double integral in polar coordinates.

(b) (Extra credit:) Evaluate the integral in part a.

3. Consider the solid S bounded by two cylinders x2 + y2 = 1 and x2 + z2 = 1. (a) Find the volume of S.

(b) Find the surface area of S

4. (Extra credit:) Consider the solid S bounded by three cylinders x2 +y2 = 1, x2 +z2 = 1, and y2 + z2 = 1.

(a) Find the volume of S.
(b) Find the surface area of S

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