2.4 Fundamental Concepts of
Integral Calculus
(Calc II Review)
Integral and Derivative Are
Complements
• Derivative: Give me distance and time,
and I’ll give you velocity (speed, rate)
• Integral: Give me velocity and time, and
I’ll give you distance
Distance = Velocity x Time ;
Area = Width x Height
• From algebra, we know that d = v t
• From geometry, we know that rectangular
area A = w h
v
d
t
h
A
w
Changing Velocity as a
Sequence of Rectangles
v3
v4
v5
v6
Total distance =
d1 + d2 + d3 + d4 +
d5 + d6
v2
v1
d1
d2
d3
d4
t1
t2
t3
t4
d5 d6
t5
t6
Estimating Area Under Points
What if instead of rectangles, we were given
points: how could we use rectangles to estimate
area under points?
Underestimating Area
• Here we underestimate the area by putting left
corners at points:
Overestimating Area
• Here we overestimate the area by putting right
corners at points:
Left- and Right-Hand Sums
• As with derivative, we can replace
• (t2-t1), (t3-t2), etc., with a general ∆t.
• v with a function f(t)
• So for n time values
• left-hand-sum (underestimate) is
f(t0)∆t + f(t1)∆t + f(t2)∆t + … + f(tn-1)∆t
• right-hand-sum (overestimate) is
f(t1)∆t + f(t2)∆t + f(t3)∆t + … + f(tn)∆t
Definite Integral
• Let’s say that t goes from a starting value a to an
ending value b.
• As ∆t gets smaller, we have more points n and a
smaller difference between left- and right-hand
sums.
• In the limit, this gives us the definite integral….
b
∫a f(t) dt =
lim (f(t0)∆t + f(t1)∆t + … + f(tn-1)∆t )
n➔∞
= lim (f(t1)∆t + f(t2)∆t + … + f(tn)∆t )
n➔∞
Total Change
b
∫a F’(t) dt =
(
total change in F(t)
from t = a to t = b
) = F(b) - F(a)
In other words: If F’ is the derivative of F, we
can compute the integral (total change) from a to
be by plugging in these values to F and taking
the difference.
Computational Science vs.
Calculus
• Calculus tells you how to compute precise
integrals & derivatives when you know the
equation (analytical form) for a problem; e.g.,
for indefinite integral:
∫(-t2 + 10t + 24) dt =
-t3
3
+ 5 t2 + 24t + C
• Computational science provides methods for
estimating integrals and derivatives from
actual data.