MATH1003 Integral Calculus & Modelling
Introduction to Integral Calculus &
Modelling
 Rates of Change: Motion
 Differential Equations
 Infinite Sequences
Integral Calculus
 The Definite Integral: Definition
 The Fundamental Theorem of Calculus
 Properties of the Definite Integral
 The Natural Logarithm
→
Definition
→
Logarithmic Differentiation
 The Natural Exponential Function
→
Definition
→
The General Exponential Function
 Growth Rates
Models and Differential Equations (DEs)
 Introduction to Models and DEs
 Volumes of Solids of Revolution
→
Mathematical Models
→
Volumes by Disks
→
Solution Curves
→
Volumes by Shells
→
Direction Fields
 Separable Equations
Integration Techniques
 Integration by Substitution
→
→
Classifying DEs
→
Solving 1st order, 1st degree separable
DEs
Variation to u-substitution
 Integration by Parts
→
Definite Integrals
→
Reduction Formulae
 Integration by Reduction to Partial
Fractions
 Integrals as Functions
 Applications of Separable Equations
→
Constructing DEs
 Linear DEs
 Second Order DEs
→
Notation
→
Linear 2 order DEs
→
Solutions
nd
st
 Systems of DEs – Reduction of 1 order
linear to 2nd order
Introduction to Integral Calculus &
Modelling
y = dependent variable
x = independent variable
Rates of Change: Motion
For a falling body, v(t) = gt.
d
1
[s(t)]= g(t) ⟹ s(t) = gt2 {by integration}
dt
2
2
Differential Equations
dP
=± kP
dt
 Exponential Growth and Decay
 Force and Acceleration
→
F(x,t) = m
For SHM:
m
d2 x
dt2
d2 x
dt2
= − kx
dx
dy
=Ax − Bxy,
= − cy +Dxy
dt
dt
 Systems of Differential Equations
 Partial Differential Equations
→
Heat Equation
∂2 T
∂T
=k
2
∂x
∂t
→
Wave Equation
∂2 y 2 ∂y2
=c
∂x2
∂t2
Infinite Sequences
 Arithmetic Sequences
e.g. 1, 4, 7, 10, …
 Geometric Sequences
e.g. 1, 2, 4, 8, 16, …
 Fibonacci Sequence
1, 1, 2, 3, 5, 8, …
 Other Sequences
e.g. 1, 2 , 3 , 4 , …
1 1 1
The terms of a general sequence of real numbers are labelled as: a0, a1, a2, …
lim an = L {the limit of a sequence}
n→∞
 If L is a constant → the sequence is convergent
 If L does not approach a finite value → the sequence is divergent
3