MATH 251 – LECTURE 30
JENS FORSGÅRD
http://www.math.tamu.edu/~jensf/
This week: 14.5–7
webAssign: 14.5–6, due 4/18 11:55 p.m.
Next week: 14.7
Friday:
webAssign: 14.7, opens 4/18 12 a.m.
Midterm 3: 14.1–7
Help Sessions:
Sun–Thu 6–8 p.m. in BLOC 149
Office Hours:
BLOC 641C
M 12:30–2:30 p.m.
W 2–3 p.m.
or by appointment.
Surface area
Exercise 1. Find the surface area of the sphere with radius R.
Surface area
Exercise 2. Find the surface area of the part of the plane x+2y +3z = 1 that lies inside the cylinder x2 +y 2 = 2.
Surface area
Exercise 3. Find the surface area of the surface with parametric equations x = u2, y = uv, z = v 2, where
0 ≤ u ≤ 2 and 0 ≤ v ≤ 3.
Surface integrals
Let S be a parametrized surface. It is natural to write
ZZ
Area(S) =
dS,
S
so that
ZZ
ZZ
|r0u × r0v | dA.
dS =
S
D
Definition 4. The integral of a function f (x, y, z) over the parametrized surface S is
ZZ
ZZ
f (x, y, z)dS =
f (r(u, v)) |r0u × r0v | dA.
S
D
Surface integrals
Exercise 5. Evaluate
RR
S
ydS, where S is the part of the plane 3x + 2y + z = 6 that lies in the positive octant.
Surface integrals
Exercise 6. Evaluate
RR p
S
1 + x2 + y 2 dS, where S is parametrized by the vector valued function
r(u, v) = hu cos(v), u sin(v), vi
for {0 ≤ u ≤ 1, 0 ≤ v ≤ π}.
Surface integrals
Exercise 7. Let S be the unit sphere with density ρ(x, y, z) = z 2 + z + 1. Compute the total mass of S.