MAT 113 In-Class Activity
5.3, 5.9: U-Substitution
3
x
dx . Clearly identify u and du.
1. Evaluate
x 1
0
Solution: Let u x 1 . Then du dx . Notice also that x u 1.
Method 1:
OR
When x 0, u 1 and when x 3, u 4 .
Method 2:
0
4
u u
1
1
2
12
du
2
u 2u
3
MTHSC 113
Section 9.4: Convergence Tests
Name: _
1. Find the nth or general term of the series. State if the series converges or diverges.
1
1
1
1
1
1
1
3 3
3
.
2
3
3
4
9
16
25
n 1
n 2 n 1 n 3
Since p
2
3
1 , by the p-series test the series
1
n
n 1
2
MAT 113
Sections 9.3: Infinite Series
Name: _
1. What is the condition for convergence of the geometric series
ar
k
?
k 0
Solution:
The series converges if and only if | |
Equivalently: -1<r<1
2. Explain why or why not: State whether the following statem
MTHSC 113
Section 9.1: Sequences
Name: _
Several terms of a sequence an n 1 are given. For each sequence, do the following: (a) find the next two terms of
the sequence, (b) Find a recurrence relation that generates the sequence (supply the initial value o
MAT 113 In-Class Activity
7.5: Integrating Rational Functions by Partial Fractions
1. Evaluate
Name: _
x2 8x 3
x3 3x2 dx .
Solution:
x2 8x 3 x2 8x 3 A B
C
2
2
3
2
x 3x
x ( x 3)
xx
x3
x 2 8 x 3 Ax( x 3) B( x 3) Cx 2 Ax 2 3 Ax Bx 3B Cx 2 ( A C ) x 2 (3 A
MAT 113 In-Class Activity
7.2: Integration by Parts
e
1.
2 x
Name: _
sin(2 x )dx
Solution:
This solution requires the use of integration by parts twice and some algebra.
Let u sin 2 x . Then du 2cos 2 x dx . Let dv e2 x dx and then v 1 e2 x .
2
(Alternati
MAT 113 In-Class Activity
6.4: Length of a Plane Curve
Name: _
1. Set up and evaluate the definite integral that gives the length of the curve x
dx 1 1/2
1
y 1/2
dy 2
2y
Solution:
9
and
L
1
y 3/ 2
y1/ 2 , from y 1 to y 9 .
3
2
1
1
1 y1/2 1/2 dy
2y
2
E
MAT 113 In-Class Activity
6.2: Volumes by Slicing: Disks and Washers
Name: _
1. Find the volume for the solid generated by revolving the region bounded by f ( x)
about the x-axis.
Solution:
2
2
1
V dx 2 dx
1
1x
x
2
2
x1
2
2
cubic units
2. Find the volu
MAT 113 In-Class Activity
6.1: Area Between Two Curves
Name: _Key_
1. Find the area of the region enclosed by the curves x y 2 and x y 2 .
a) Sketch the enclosed region. Label the curves and points of intersection.
Solution: The points of intersection are
MthSc 108 Learning Activity
Section 9.5: The Ratio and Comparison Tests
Name: _KEY_
1
converges or diverges.
n
n 1
1
1 1
Solution: Compare this series to the convergent p-series, 2 2 , where p 2 1 .
2 n 1 n
n 1 2n
1
2
a
2n 2
lim n lim 2n n lim 2
1, where