Integral Calculus for Engineering and other Hard Sciences
MATH 106

Fall 2012
Cylindrical Shells
Cylindrical Shells
Consider rotating the region between the curve
y = x2
the line
x=2
and the xaxis about the yaxis.
If instead of taking a cross section perpendicular to the yaxis, we take a cross section
perpendicular to the xaxis
Integral Calculus for Engineering and other Hard Sciences
MATH 106

Fall 2012
MATH 106 PRACTICE FINAL
Please work out each of the given problems. Credit will be based on the steps that you show
towards the final answer. Show your work.
PROBLEM 1
Evaluate the given integrals, derivatives and limits.
A) (20 Points)
Solution
We integr
Integral Calculus for Engineering and other Hard Sciences
MATH 106

Fall 2012
Name
MATH 106 PRACTICE MIDTERM 1
Please work out each of the given problems. Credit will be based on the steps
that you show towards the final answer. Show your work.
PROBLEM 1 Please answer the following true or false. If false, explain why or provide a
Integral Calculus for Engineering and other Hard Sciences
MATH 106

Fall 2012
Partial Fractions
The Fundamental Theorem of Algebra
The fundamental theorem of algebra states that if P(x) is a polynomial of degree n then P(x) can
be factored into linear factors over the complex numbers. Furthermore, P(x) can be factored
over the real
Integral Calculus for Engineering and other Hard Sciences
MATH 106

Fall 2012
Applying Hookes Law to Rubber Bands
Hookes law goes back to the 17 th century when English physicist Robert Hooke
discovered that elastic objects such as springs stretch in proportion to the force that acts on them.
Up to this day this law holds true[1] a
Integral Calculus for Engineering and other Hard Sciences
MATH 106

Fall 2012
Arclength and Surface Area
Arclength
Example
Suppose that you want to design a ski lift and need to determine how much cable will be needed
between two posts that are 66 feet apart. The formula for the cable is
y = 50 cosh(x/50)
To compute the length of t
Integral Calculus for Engineering and other Hard Sciences
MATH 106

Fall 2012
MATH 106
CALCULUS and ANALYTIC GEOMETRY
Mon, Wed, and Fri 9:25 to 10:55 AM
Room A 211
5 UNITS
Instructor Larry Green
Phone Number Office: 5414660 Extension 341
email:.[email protected]
Class Grades
Web Page: http:/www.ltcconline.net/greenl/courses/106/106
Integral Calculus for Engineering and other Hard Sciences
MATH 106

Fall 2012
Area Between Two Curves
Area Between Two Curves
Recall that the area under a curve and above the x axis can be computed by the definite integral.
If we have two curves
y = f(x) and y = g(x)
such that
f(x) > g(x)
then the area between them bounded by
the h
Integral Calculus for Engineering and other Hard Sciences
MATH 106

Fall 2012
Trigonometric Substitution
Integration Involving
When we have integrals that involve the above square root, we can use the appropriate
substitution.
Example:
We have
Exercises
A.
B.
C.
Two Key Formulas
From Trigonometry, we have the following two key form
Integral Calculus for Engineering and other Hard Sciences
MATH 106

Fall 2012
NAME
MATH 106 PRACTICE MIDTERM 2
Please work out each of the given problems. Credit will be based on the steps
that you show towards the final answer. Show your work.
PROBLEM 1 Set up the integrals that solve the following problems. Sketch the appropriate
Integral Calculus for Engineering and other Hard Sciences
MATH 106

Fall 2012
MATH 106 MIDTERM 3
Please work out each of the given problems. Credit will be based on the steps
that you show towards the final answer. Show your work.
PROBLEM 1
You are constructing the newest resort at the bottom of Lake Tahoe (which is 1654 feet
deep)