Fall 2010 Math 152
2
Section 8.1
Week in Review IV
Key Points:
courtesy: David J. Manuel
(covering 7.5, 8.1, and 8.2)
1.
1
ˆ
u dv = uv −
ˆ
v du
Section 7.5
Key Points:
1. favg
1
=
b−a
Examples:
ˆ
b
f (x) dx
a
1. Compute the following integrals:
2. If f > 0, favg is the height of the rectangle
whose area equals the area under the graph
of f from x = a to x = b. (NOTE: if f is
continuous, the rectangle will always intersect
the graph of f ).
(a)
ˆ
x2 sin(3x) dx
(b)
ˆ
y 3 e−y dy
(c)
ˆ
Examples:
1/2
sin−1 (2x) dx
0
1. Find the average value of the given function
over the given interval:
2
(a) f (x) = √ over [1, 4]
x x
√
(b) f (x) = x x2 + a2 over [0, a]
(c) f (x) =
2
(d)
ˆ
(e)
ˆ
1
1
over [−2, 1]
4 − 3x
(f)
e2x − 1
over [0, 3]
ex
h πi
(e) f (x) = sin3 x cos x over 0,
2
ˆ
t3
e
p
1 − t2 dt
ln x
dx
x2
e3x cos(2x) dx
(d) f (x) =
(f) f (x) =
2. Find the volume obtained by rotating the region bounded by y = sin x, y = cos x, x =
π
0, and x = about the y-axis.
4
1
over [−1, 1]
1 + x2
1
3
Section 8.2
Key Points:
1. Know the integrals of all six trig functions
2. Know identities to integrate the squares of all
six trig functions
3. Other trig integrals: when possible, “save” a
du and rewrite the rest of the integral in terms
of u using identities
4. Product to Sum integrals: Based on the Angle
Addition identities
Examples:
1. Integrate the following:
ˆ
(a)
cos3 (2x) dx
(b)
ˆ
(c)
ˆ0
(d)
ˆ
π/2
√
sin3 x cos x dx
sec3 x tan3 x dx
π/2
csc3/2 x cot3 x dx
π/6
(e)
ˆ
sin(6x) sin(4x) dx
(f)
ˆ
sin(2x) cos(4x) dx
2. Find the volume of the solid obtained by rotating the region under the graph of y =
sin2 x and above the x axis (one period only)
about the x-axis.
2