Transcript Areas Between Curves

A Quick Review of Math 200
 Calculus
rests on 3 critical ideas/concepts:
Limit
Instantaneous
rates of
change
Total accumulated
change
Rate of Change…
 Relates
to slope in the real world
 By using limit the “Rise/Run” becomes the
Newton Quotient
 And the rest…



Max/Min
Related rates
Differential Equations
Perhaps one of the most important
graphs in history!
400
(2010,388)
390
CO2 Concentration (ppmV)
380
370
Slope is about
1.4 ppm/year
360
350
340
330
320
310
(1958,315)
300
1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015
Year
Snapshot of Earth Breathing!
Annual Variation of Atmospheric CO2
(2008 data)
389
May
388
June
April
CO2 Concentration
387
July
386
Jan
Feb
Mar
Dec
385
384
383
382
August
Sept Oct
Nov
Total Accumulated Change…

Relates to the idea of area in the real world
Example: a drug is released into a
patients blood stream at a rate
given by
2
r (t )  4t  t
mg/minute. How much drug does
the patient receive in 4 minutes?
Units are mg/min
Working with Integrals…
 Pay



attention to 5.2 and 5.4 in text
Substitution: simplest of the methods (Chp
7.1)
Integration by parts (coming in Chp 7)
etc
Which of the following are good
candidates for substitution?
A)
 tan x dx
sin x
B) 
dx
cos x
x
C )  xe dx
D)
 (x
2
3
 1)(x  x )dx
Areas Between Curves
Section 6.1
Some warm ups…
 What
is the area between the x-axis and
the graph of
3
f ( x)  x  x
 When
is an area “equal to” an integral?
 What’s
the area between the x-axis, the
lines x = 0 and x = 2 and the graph of the
function
3
f ( x)  x
 What’s
the area between the y-axis and
the above function (x = 0 and x = 2)?
 What’s the area between the curves y = x3
and y = x?
Can we develop a strategy?
 Checklist
of things to watch for when
finding areas between curves…
Some examples…
 Pg
331: 68,71,72