Math 136: Calculus 2
Continuous Income Streams
Professor Levandosky
1. Suppose each of the following continuous income streams is invested in an account that
earns 3% annual interest, compounded continuously. Calculate the following for each
income stream
Math 136: Calculus 2
Fall 2012, Professor Levandosky
Geometric Series Worksheet
Recall the following basic facts about geometric series.
The sum of a nite geometric series is
n
ark = a + ar + ar2 + + arn =
k=0
a(1 rn+1 )
1r
The sum of an innite geometri
Math 136: Calculus 2
Worksheet 3, Fall 2012
Professor Levandosky
The center of mass of a planar region with constant mass density that lies below the
graph of a function f (x) over an interval [a, b] has coordinates (x, y ) given by
b
xf (x) dx
a
x = b
a
Math 136: Calculus 2
Worksheet 2, Fall 2012
Professor Levandosky
1. For each improper integral, (a) identify the point or points where the integral is improper and (b) evaluate the integral.
/2
sec2 (x) dx
(a)
0
1
(b)
1
1
1
dx
1 x2
ln(x) dx
(c)
0
2. Supp
Math 136: Calculus 2
Spring 2012
Worksheet 4: Center of Mass
Professor Levandosky
The center of mass of a planar region with constant mass density that lies below the
graph of a function f (x) over an interval [a, b] has coordinates (x, y ) given by
b
xf
Math 136: Calculus 2
Spring 2012
Worksheet 3: Improper Integrals
Professor Levandosky
1
1
dx in each of the following cases. Either show that the integral diverges,
p
0x
or show that it converges and nd its value.
1. Evaluate
(a) p < 1
(b) p = 1
(c) p > 1
College of the Holy Cross, Fall Semester, 2009
Math 136
Group Worksheet 3
For each of the following series, ll in the entries of the table provided. Record the rst
six digits after the decimal point, but leave the entire number in the calculator for use i
Math 136: Calculus 2
Spring 2012
Worksheet 1
Professor Levandosky
1. Steve was curious to know just how much snow he shoveled from his driveway one
winter. So one day he went out and took measurements. The driveway is approximately 115 feet long, and the
Quiz 4 Solutions (Section 1) - Math 136 - Fall 2012
1
ln(1 + x2 ) dx.
1. Consider the denite integral
0
(a) Use Simpsons Rule with n = 4 subintervals to approximate the integral. Round
the approximation to 5 decimal places.
Solution. x = 1/4, so the Simps
Quiz 4 Solutions (Section 4) - Math 136 - Fall 2012
2
[ln(x)]2 dx.
1. Consider the denite integral
1
(a) Use Simpsons Rule with n = 4 subintervals to approximate the integral. Round
the approximation to 5 decimal places.
Solution. x = 1/4, so the Simpsons
Quiz 3 Solutions (Section 1) - Math 136 - Fall 2012
Evaluate each integral using the indicated method.
1.
x2 ex dx (parts)
Solution. First let u = x2 , dv = ex dx. This gives du = 2xdx and v = ex , so
2 x
2 x
x e dx = x e + 2xex dx
In the new integral, le
Quiz 3 Solutions (Section 4) - Math 136 - Fall 2012
Evaluate each integral using the indicated method.
1.
x2 cos(x) dx (parts)
Solution. First let u = x2 , dv = cos(x) dx. This gives du = 2xdx and v = sin(x), so
2
2
x cos(x) dx = x sin(x) 2x sin(x) dx
In
Math 136, Midterm Exam 3 Solutions
Prof. Levandosky
1. Let R denote the region in the rst quadrant bounded by the curve y = ex and the
lines y = 0, x = 0 and x = 1. Find the coordinates (, y ) of the center of mass of R.
x
Round your answers to 3 decimal
Math 136, Midterm Exam 2 Solutions
Prof. Levandosky
1. Use a trig substitution to evaluate the following denite integral.
2
4 x2 dx
0
Solution. Let x = 2 sin , dx = 2 cos d. Then 4 x2 = 2 cos , so the integral
becomes
x=2
4 cos2 d
x=0
The limits of int
Math 136, Midterm Exam 1 Solutions
Prof. Levandosky
1. Suppose an object begins from rest and its acceleration at time t is given by a(t) =
4t + sin(t).
(a) Find the velocity of the object at time t.
Solution.
v (t) = 4t + sin(t) dt = 2t2 cos(t) + C
Since