2'4 Fundamental Concepts of Integral Calculus Calc II Review

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2.4 Fundamental Concepts of Integral
Calculus(Calc II Review)
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Integral and Derivative Are Complements

• Derivative Give me distance and time, and Ill
  give you velocity (speed, rate)
• Integral Give me velocity and time, and Ill
  give you distance

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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

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Changing Velocity as a Sequence of Rectangles
v6
v5
v4
v3
Total distance d1 d2 d3 d4 d5 d6
v2
v1
d1
d2
d3
d4
d5
d6
t1
t2
t3
t4
t5
t6
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Estimating Area Under Points
What if instead of rectangles, we were given
points how could we use rectangles to estimate
area under points?
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Underestimating Area

• Here we underestimate the area by putting left
  corners at points

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Overestimating Area

• Here we overestimate the area by putting right
  corners at points

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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

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Definite Integral

• Lets 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

• ? f(t) dt lim (f(t0)?t f(t1)?t
  f(tn-1)?t )

a
n ? 8
lim (f(t1)?t f(t2)?t f(tn)?t )
n ? 8
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Total Change
total change in F(t) from t a to t b
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.
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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 5 t2 24t
  C
• Computational science provides methods for
  estimating integrals and derivatives from actual
  data.

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