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

d

t

v A

w

h

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

Estimating Area Under PointsWhat 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….

∫ 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 ∞ ➔

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.

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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