Logarithmic and
Exponential
Functions
Objectives:
You should be able to
contrast differentiation and
integration formulas of
logarithmic and exponential
functions
evaluate integrals involving
logarithmic and exponential
functions
Integration of Logarith
General Power
Formula
Objectives:
You should be able to
derive the general power
formula
identify integrals that can
be solved using the general
power formula
evaluate integrals using
general power formula
The General Power Formula:
Example:
Example:
E
Trigonometric
Substitution
Integrals involving square roots such as
where a is a constant can be solved using
trigonometric substitutions. These substitutions are
as follows:
x = a sin for integrals with
x = a tan for integrals with
x = a sec for integral
Inverse
Trigonometric
Functions
Objectives:
You should be able to
contrast differentiation and
integration formulas of
inverse trigonometric
functions
evaluate integrals involving
inverse trigonometric
functions
Formulas:
Example:
Example:
Example:
Exam
Integration Process
In many Calculus references,
the lists containing integration
formulas are extensive. Some
contain more than fifty
formulas while others have
more than a hundred. Since
recalling formula is difficult,
the following integration
procedur
Solvable
Groups of
Trigonometric
Integrals
Group I: Integrals of the form
where m and/or n are positive
integers
Case 1: When m is odd factor out sin u du and then
change the
remaining sine into cosine using the
identity
sin2 u 1 cos2 u
Simplify and integ
Integration by
Substitution:
Trigonometric
Functions
Objectives:
You should be able to:
contrast differentiation and integration
formulas of trigonometric functions
evaluate integrals involving
trigonometric functions
Formulas:
cosudu sinu C
2
csc
udu co
Evaluating Definite
Integrals
Now that you already know how to
evaluate indefinite integrals, evaluation
of definite integrals becomes very easy!
All you have to do is drop the constant
b
C of the indefinite integral
and affix the
a
symbol for evaluation
A Special Technique of
Integration
When
the integrand is a
fraction of the form ,
where P(x) and Q(x) are
polynomials in x, and when
the degree of P(x) is greater
than or equal to the degree
of Q(x), then divide P(x) by
Q(x) first before using any
formula
1. Methane and oxygen react in the presence of a catalyst to form formaldehyde.
In a parallel reaction, methane is oxidized to carbon dioxide and water:
CH4 + O2 HCHO + H2O
CH4 + 2O2 CO2 + 2 H2O
The feed to the reactor contains equimolar amounts of methan
3000 kg of 10% NaNO3 solution is available. It is
desired that 95% of the salt be recovered as
crystals. This is done by cooling to 10C ( solubility
= 5lg/100 kg water) Calculate (a) weight of
crystals (b) weight of water that must be removed
from the or
Purge
Purge or bleed is a flow arrangement
used in recycling operations where a
fraction of the recycle stream
becomes an output stream.
Purge
In recycling operations, a separator removes
only the desired product .
The inerts and the impurities which ente
1/17/2013
Review: Material Balance
with Chemical Reaction
Fractional conversion
Part of the reactant that is converted into
product
=
=
Review: Material Balance with Chemical Reaction
Review: Material Balance with Chemical Reaction
200 moles of SO2
RECYCLE
A system with recycle is one in
which a stream leaving a
processing unit is partly
returned to the same unit for
reprocessing or reusing.
RECYCLE
increase yield
increase purity
enrich a product
conserve heat
improve operations
RECYCLE
Recycle str
ANSwE/i KEY 411/023 L92
- _ A 4 #
MATH 023 Lin] 2 a _ 3
Us: on: whale yellow papigulrizad lengthwise, (50 mum'les Duly) D ) l 7 f0 534 ' 33) E g f 7 1,514? 043,
Forms,1m2,cansidet1nxeginus1,R1,andean 1 . r
am right
1. Find the moment of mania I, and II Df
WGRK DUNE IN PUWIPWGDUT
IJQUDJS
Please make the Izlecessarjyr correction in your
notes.
Re: correct bounds for the set-up (with x-
axis on top) should he 4 to 8
S
W = 9800].:(4shv = 2.96xlDJ
4
Center of Mass (Centroid)
Regions Bounded by Curves:
For horizontal subdivisions:
For vertical subdivisions:
Example 1
Find the centroid
of a thin
triangular plate
bounded by the
curves y=2x,
x=1, and y=0.
Example 2
Find the
centroid of the
region
bounded
Moment of Inertia
The strength of structural members depends to a large extent on the
properties of their cross sections, particularly on the second moments, or
moments of inertia, of their areas.
Moments of inertia are used in courses such as mechanics o
WORK PROBLEMS
First recall the definition of force from Newtons Law that
Force = mass x acceleration
In symbols,
F = ma
Work is a technical term meaning the product of force F and the
displacement caused by it.
In symbols,
W=Fd
Total Work Done
The total w
Total Force Exerted by
Liquids
Application:
Finding the total force caused by liquid pressure on a plate
submerged on the liquid or on the side of the container holding the
liquid.
Assumption:
The liquid is in an open container, or there is a free surface
INDUSTRIAL MATERIAL and PROCESSES
(LABORATORY)
QUIZ# 3P and 6P
ASSIGNMENT
Name:
Score:
Section: 3IE3
Date: 07-28-17
1.) How to use a Vernier caliper?
- Clean the object you are measuring.
- Unlock the screw.
- Slide one of the jaws against the object.
- R
Dist-012H
Revised: Nov 19, 2012
Extractive Distillation for Heptane-Toluene Separation using
Aspen HYSYS V8.0
1. Lesson Objectives
Essentials of extractive distillation
How to compare design alternatives
2. Prerequisites
Aspen HYSYS V8.0
Introduction to d
Revised: Nov 30, 2012
Get Started Guide for New Users of Aspen HYSYS V8.0
1. Lesson Objectives
Learn how to create a new Aspen HYSYS simulation
Learn to construct flowsheet, including adding blocks and streams, reconnecting streams, and
breaking/joining s
Dist-009H
Revised: Nov 19, 2012
Separation of Acetone-Water with Aspen HYSYS V8.0
Liquid-Liquid Extraction with 3-Methylhexane as the Solvent
1. Lesson Objectives
Learn how to build an extraction and solvent recovery flowsheet.
Learn how to configure a li
=
=
= ( )
We shall set up Stoichiometry Tables using
species A as our basis of calculation in the
following reaction. We will use the
stochiometric tables to express the
concentration as a function of conversion. We
will combine Ci = f(X) w