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MATH 20E
FINAL
Please answer the following questions. You will not get credit for answers un- less you demonstrate how you arrived at them. In short, please show all work. Numerical integrations by calculator will not be accepted.
Problem #1 ( pts.) Consider a wire which is shaped like the 3D cork-screw curve ~c(t) =
cos(t), sin(t), t
for 0 6 t 6 π. Suppose the mass density is given by f (x, y, z) = 2 + x^2 + xy (in units of mass per units of length). Please compute the mass of this wire. (Hint: This is just a path integral problem.)
1
Problem #2 ( pts. total) Please compute the surface area of the portion of the graph of z = f (x, y) = 1 − x^2 − y^2 which lies over the unit disk x^2 + y^2 6 1.
Problem #4 ( pts. total) Consider the diamond in the plane R = {− 4 6 x + y 6 4 , − 4 6 x − y 6 4 } (see the picture below). Use the change of variables x = u + v, y = v − u to compute the integral:
I =
R
(1 + x + y) dxdy.
(Hint: One of the main things you need to figure out is the limits of integration in the (u, v) coordinates. It helps to notice that the domain in the (u, v) coordinates will be a box −a 6 u 6 a and −a 6 v 6 a. You just need to find the correct value of a.)
Problem #5 ( pts. total)
Compute the outward flux
S F~ · n dSˆ of the vector-field F~ = (x^3 , y^3 , z) over the entire surface of the cylinder S = {x^2 + y^2 6 1 , z = 0} ∪ {x^2 + y^2 = 1, 0 6 z 6 1 } ∪ {x^2 + y^2 6 1 , z = 1}.
