Two-dimensional applications of definite integrals.

1. Integral Calculus
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2. Integral Calculus ­ Who needs it!
(What is it good for?)
Examples from physics:
Displacement = Velocity x Time
1. If you are traveling at 60mph, how far do you travel in 2 hours?
Sketch a velocity­time graph to illustrate this distance.
2. From a stop light, you accelerate at a constant rate to 60 mph in 10 seconds.
How far have you traveled? Sketch a velocity­time graph to illustrate this distance.
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3. Work = Force x Displacement
1. How much work is done by a weightlifter to raise a 500 lb. barbell 8 ft. in the air ?
Sketch a force­displacement graph to illustrate the work done by the weightlifter.
Use the vertical axis for force and the horizontal axis as displacement.
Note: The force due to the gravitational pull of the earth is defined to be the weight of an object: Fgrav = weight.
What do these 3 examples have in common?
(Why is there no need for integral calculus to answer these 3 questions?)
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4. Section 5­8c: Applications of the Definite Integral
Integral calculus provides a way to find a _____________ in which one factor _________.
(Graphically, the area under a function cannot be determined by a simple geometric formula. )
In these cases we add up a whole bunch (infinitely many) of
really small (infinitesimal) bits of area to find the total area.
f
(x, f(x))
dx
a b
By the Fundamental Theorem of Calculus
If: f is an integrable function on the interval [a, b].
1. _____________________________________
f(x) = g ' (x)
2. _____________________________________
Then:
Graphical meaning: Area
__________________ = Height
_____________ Width
_____________
by FTC hypothsis: derivative of g
instantaneous
by definition rate of change
of derivative
of g infinitesimal interval
and dx:
Let's put some clothes on this "naked math"
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5. Let's put some clothes on this "naked math"
In applications of the definite integral, the area under the function in the interval [a, b] has a physical meaning,
so the area of each strip (dA) has a physical meaning.
f
?
Height means___ Instantaneous Rate of change of ___ ?
?
dx means ___ Infitesimal interval of ___ ?
a b
?
dA means __
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6. Displacement = velocity x time
If the velocity throughout the time interval dt
isessentially constant at v(t), and the length
of time is dt, then D = v(t) x dt
Both f(x) and dx have physical meanings.
The dx is the infinitesimal time interval (base of rectangle),
and f(x) is the velocity during that time interval (height of
rectangle.)
Any Riemann sum Rn will be bounded above by
the Upper sum, Un and bounded below by the
lower sum, Ln: Ln ≤ Rn ≤ Un. Taking the limit
of Ln and Un, as the number of rectangles increases
without bound, Rn will be squeezed between
the limits of Ln and Un.
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7. Work is the product of force and displacement.
The definite integral is used for work because in
the product, one of the factors ­ force varies with
respect to displacment.
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