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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 I’ll give you velocity (speed, rate)
• Integral: Give me velocity and time, and I’ll 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
d
t
v A
w
h
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Changing Velocity as a Sequence of Rectangles
d1
t1
v1
d2
t2
v2
d3
t3
v3
d4
t4
v4
d5
t5
v5
t6
v6
d6
Total distance = d1 + d2 + d3 + d4 + d5 + d6
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Estimating Area Under PointsWhat 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• 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….
∫ f(t) dt = lim (f(t0)∆t + f(t1)∆t + … + f(tn-1)∆t )a
b
= lim (f(t1)∆t + f(t2)∆t + … + f(tn)∆t )n ∞ ➔
n ∞ ➔
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Total Change
∫ F’(t) dt = ( ) = F(b) - F(a)a
b 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.
-t3
3