Math 1B
Exam 2 Review
The in class portion of the second exam will be on Friday, April 1st. It will cover sections 7.1 – 7.5, 7.8,
8.1, and 8.2. You are allowed to use one 3” by 5” notecard (no specific examples, no unit circle, no table
of integrals). Prepare well and good luck!
You will need to be able to:
1.
2.
3.
4.
5.
Evaluate definite and indefinite integrals with integration by parts, trig integrals, trig substitutions,
partial fractions, rationalizing substitutions, etc.
Evaluate improper integrals.
Find volumes of solids of revolution using disks, washers, and shells.
Find the arc length of a curve.
Find the surface area of a solid of revolution.
Check out the Chapter 7 Review Exercises, pages 530-532, # 1-27 odd, 31-37 odd, 41-49 odd, 75, 76
Additional Problems:
1.
Evaluate each integral.
a) ∫ sin 3 x cos4 x dx
2.
3.
4.
5.
€
€
d)
∫x e
g)
€−1
tan
∫ x dx
2 −x
dx
∫
e)
∫x
€
h)
€
ln x
dx
x2
b)
∫
3
x 2 −1
∫1 x dx
5x −1
dx
f) ∫ 2
x − 2x −15
€ 2x −1
dx
i) ∫
2x + 3
€
c)
9 − x 2 dx
1
2
€
dx
x + 10x + 26
€
€4
€
x − x 3 − x −1 €
5
3
dx
j) ∫ tan x sec x dx
k) ∫
x3 − x2
€
€
€
€
€
Find the area under the curve y = 64 − x 2 between x = 0 and x = 4 .
€
€
€
The region bounded by the curve y = ln x , the line x = e , and the x-axis is revolved about the yaxis. Find the volume of the solid that is generated.
€
€
€
1
The region in the first quadrant
under the€curve y = 2
between x = 0 and x = 2 is
€
x + 4x + 3
rotated about the y-axis. Find the volume of the solid that is generated.
2
€
€that are convergent.
Determine whether each integral is convergent or divergent. Evaluate
those
€
∞
4
1
1
dx
dx
a) ∫
c) ∫
3
2
1
0
(2x + 1)
( x −1)
∞
1
1
1
dx
dx
b) ∫−∞ 2
d) ∫ 0
2
x +9
1−
x
€
€
€
€
€
€
6.
Find the length of the curve.
!
a) 𝑦 = ! 𝑥 ! + 4
7.
!
!, 0
≤𝑥≤3
b) 𝑦 = 2 ln sin
!
!
𝑥
!
,! ≤ 𝑥 ≤ 𝜋
⎛π
⎞
The portion of the curve y = tan x between (0,0) and ⎜ , 3 ⎟ is rotated around the x-axis. Set
⎝3
⎠
up, but do not evaluate, an integral to find the resulting surface area:
a) Integrating with respect to x
b) Integrating with respect to y
8.
The portion of the curve y = x 3 from x = 1 to x = 2 is rotated around the x-axis. Find the surface
area of the solid that is generated.
9.
The portion of the curve y = 1 − x 2 from x = 0 to x = 1 is rotated around the y-axis. Find the
surface area of the solid that is generated.
Answers to Additional Problems
€
€
1
1
cos7 x − cos 5 x + C
7
5
1
1
b) − ln x − + C
x
x
π
c) 3 −
3
2 −x
d) −x e − 2xe−x − 2e−x + C
52
32
1
e) (9 − x 2 ) − 3(9 − x 2 ) + C
5
f) 2ln x + 3 + 3ln x − 5 + C
1
g) x tan−1 x − ln(1+ x 2 ) + C
2
2
h) ln x + 10x + 26 + x + 5 + C
€
i)
1.
€
€
€
€
€
€
€
2.
€
€
a)
# 2x −1 &
2x −1 − 2tan−1%
(+C
$ 2 '
1
2
1
j) sec 7 x − sec 5 x + sec 3 x + C
7
5
3
1
1
x
+C
k) x 2 − + 2ln
2
x
x −1
16π
+8 3
3
3.
π 2
(e + 1)
2
4.
π (3ln 5 − 4 ln3)
5.
a)
1
36
6.
a)
15
2
€
€
7.
a) 𝑆 =
b) 𝑆 =
8.
9.
!
!"
!
!
b)
π
3
c) Divergent
d)
b) −2 ln 2 − 3
€
!
!
!
2𝜋 tan 𝑥 1 + sec ! 𝑥 𝑑𝑥
!
!
€
2𝜋𝑦
! ! !!! ! !!
! ! !!! ! !!
145 145 − 10 10
(5 5 − 1)
𝑑𝑦
π
2