CRMS Calculus 2010 May 17, 2010
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Transcript of CRMS Calculus 2010 May 17, 2010
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Integral Calculus
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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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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.
What do these 3 examples have in common? (Why is there no need for integral calculus to answer these 3 questions?)
Note: The force due to the gravitational pull of the earth is defined to be the weight of an object: Fgrav = weight.
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Section 58c: Applications of the Definite Integral
f
a b
By the Fundamental Theorem of Calculus
If: 1. _____________________________________
2. _____________________________________
Graphical meaning: _____________ _______________________________ =
Then:
by FTC hypothsis:
by definitionof derivativeand dx:
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.
Let's put some clothes on this "naked math"
f is an integrable function on the interval [a, b].
f(x) = g ' (x)
Area Height Width
derivative of g
instantaneousrate of change
of g infinitesimal interval
dx
(x, f(x))
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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.
Let's put some clothes on this "naked math"
f
a b
Height means___? Instantaneous Rate of change of ___?
Infitesimal interval of ___?dx means ___?
dA means __ ?
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Displacement = velocity x time
If the velocity throughout the time interval dtisessentially constant at v(t), and the lengthof time is dt, then D = v(t) x dt
Any Riemann sum Rn will be bounded above bythe Upper sum, Un and bounded below by the lower sum, Ln: Ln ≤ Rn ≤ Un. Taking the limitof Ln and Un, as the number of rectangles increaseswithout bound, Rn will be squeezed betweenthe limits of Ln and Un.
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.)
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Work is the product of force and displacement.The definite integral is used for work because inthe product, one of the factors force varies withrespect to displacment.