MATH 251 – LECTURE 21
JENS FORSGÅRD
http://www.math.tamu.edu/~jensf/
This week: 13.9–10
webAssign: 13.8–10, due 3/21 11:55 p.m.
M W: Kevin
F: no lecture
After spring break:
webAssign: nope
M: Review chapter 13
W: Midterm 2
Help Sessions:
M W 5.30–8 p.m. in BLOC 161
Office Hours:
BLOC 641C
M 12:30–2:30 p.m.
W 2–3 p.m.
or by appointment.
Spherical coordinates
Spherical coordinates is the three dimensional analogue of polar coordinates

 x = r cos(θ) sin(ϕ)
y = r sin(θ) sin(ϕ)
 z = r cos(ϕ)
Spherical coordinates

 x = r cos(θ) sin(ϕ)
y = r sin(θ) sin(ϕ)
 z = r cos(ϕ)
Spherical coordinates
Exercise 1. Sketch the solid whose volume is given by the integral
R π/4 R π R 4
0
0
0
r2 sin(ϕ) dr dθ dϕ.
Spherical coordinates
p
Exercise 2. Find the volume V of the solid E that lies above the cone z = x2 + y 2 and below the sphere
x2 + y 2 + z 2 = 36.
Spherical coordinates
p
Exercise 2. Find the volume V of the solid E that lies above the cone z = x2 + y 2 and below the sphere
x2 + y 2 + z 2 = 36.
Spherical coordinates
Exercise 3. Let S be the unit ball {x2 + y 2 + x2 ≤ 1} with a density that is twice the square of the distance
from the z-axis. Compute the total mass of S.
Spherical coordinates
Exercise 3. Let S be the unit ball {x2 + y 2 + x2 ≤ 1} with a density that is twice the square of the distance
from the z-axis. Compute the total mass of S.
Spherical coordinates
Exercise 4. Find the volume V of the solid E that lies inside the sphere {x2 + y 2 + z 2 ≤ 1} but outside of the
two solid cylinders {(x − 12 )2 + y 2 ≤ 14 } and {(x + 12 )2 + y 2 ≤ 14 }.