5.4
Area
Between Curves and Average Value
In this
section the focus is on setting up the appropriate interval. Some of the integrals may be evaluated with
the calculator.
The Area Between Two Curves
If f(x) and g(x) are continuous with f(x) > g(x) on the interval a < x < b, then the area between curves f(x) and g(x) is given by
Transparency - #2 and
#4
#6 Sketch the region R bounded by the curves
, 
, and the lines x = 1 and
x = 3 and then find the area.
#12 Sketch the region R between by the curve 
and the line 
and then find the
area.
#10 Sketch the region R bounded by the curve 
and the lines 
and then find the
area. (This one also requires addition)
#16 Sketch the region R which is the rectangle with vertices
(1,0), (-2,0), (-2,5) and (1,5) and then find the area.
*You might check the answer with a geometry formula!!*
#8
R is the regions bounded by the curves 
, and the line x =
ln 2.
Average Value of a
Function
Let f(x) be a
function that is continuous on the interval a
< x < b. Then
the average value V of f(x)
over the interval is given by the definite integral 
#24 Find the average value of 
#28 Find the average value of 
. Sketch the graph of
the function along with the rectangle whose base is the given interval and
whose height is the average value V.
Application
#40 Records indicate that t hours past midnight, the temperature at
the local airport was 
degrees Celsius. What was the average temperature at the
airport between
Tom invests $5,000 in an account that pays 6% compounded
continuously. What is the average value
of his investment over the next 10 years?
#52 an advertising firm is hired to promote a new television
show for 3 weeks before its debut and 2 weeks afterward. After t
weeks of the advertising campaign, it is found the P(t) percent of the viewing public is aware of the show, where 
a.
What
is the average percentage of the viewing public that is aware of the show
during the 5 weeks of the advertising campaign?
b.
At
what time during the first 5 weeks of the campaign is the percentage of viewers
the same as the average percentage found in part a)?
Assignment: 5.4 page
427, 1-13 odd, 17-27 odd, 35, 39, 47-41 odd, 53. The calculator may be used to evaluate the
integrals on all problems except #21, #23, #47, and #49. Evaluate these using calculus techniques (SUBSTITUTION!!)