Section 5.1
Suppose that the curves y = f (x) and y = g(x) bound a region R on f (x) ≤ g(x). Then the area of R is given by
Z
b
g(x) − f (x) dx
A(R) =
a
where a and b are determined by solving the equation f (x) = g(x).
Section 5.2,3
Make sure you understand how to find the volume of a solid generated by revolving the curve about the x-axis or
y-axis.
Review the difference between the method of washers and shells.
Remember the volume of a disk of radius r and thickness h is πr2 h.
Also the volume of a washer of inner radius r, outer radius R, and thickness h is π(R2 − r2 )h.
For the method of shells, the volume ∆V of a thin cylindrical shell of radius x, height f (x), and thickness ∆x is
given by ∆V = 2πxf (x)∆x.
Section 5.4
The arc length of a curve given parametrically by x = f (t), y = g(t), a ≤ x ≤ b:
s
Z a 2 2
dx
dy
+
dt
L=
dt
dt
b
If your curve is given by y = f (x) and a ≤ x ≤ b, then
s
2
Z a
dy
L=
1+
dx
dx
b
If your curve is given parametrically by x = f (t), y = g(t), a ≤ x ≤ b, then the surface area is
Z a
q
2
2
A = 2π
g(t) [f 0 (t)] + [g 0 (t)] dt
b
If your curve is given by y = f (x) and a ≤ x ≤ b, then
Z a
q
2
A = 2π
f (x) 1 + [f 0 (t)] dx
b
Section 5.5
The work done by a force F in moving an object along a straight line from a to b is W = F · (b − a) if F is constant,
but is
Z
b
W =
F (x) dx
a
if F = F (x) is a variable.
Make sure to review problems like those on examples 2, 3, 4 and 5.
Section 5.6
Given density δ(x), the center of mass x is the total moment M over mass m:
x=
Rb
xδ(x) dx
M
= Ra b
m
δ(x) dx
a
Section 7.1
I do not want to type all the formulas you should know, so just go over the formulas on pages 383 and 384. This
section basically deals with different ways to use u-substitution.
Section 7.2
The integration-by-parts formula says that
Z
Z
u dv = uv −
v du
Section 7.3
For this section, make sure you go over the following types of trig integrals:
R
R
1. sinn (x) dx and cosn (x) dx
R
2. sinn (x) cosm (x) dx
R
R
R
3. sin(n x) cos(m x) dx, sin(n x) sin(m x) dx, cos(n x) cos(m x) dx
R
R
4. tann (x) dx, cotn (x) dx
R
R
5. tann (x) secm (x) dx, cotn (x) cscm (x) dx
Some trig identities are useful!
sin2 x + cos2 x = 1
1 + tan2 x = sec2 x
1 + cot2 x = csc2 x
2 sin2 x = 1 − cos(2x)
2 cos2 x = 1 + cos(2x)
2 sin(mx) cos(nx) = sin (m + n)x + sin (m − n)x
2 sin(mx) sin(nx) = cos (m − n)x + cos (m + n)x
2 cos(mx) cos(nx) = cos (m + n)x + cos (m − n)x