Review 6 • Riemann Sums and Trapezoidal Rule • Definite integrals (limits)
Review 6
Jan 15, 2016
Review 6. Riemann Sums and Trapezoidal Rule Definite integrals (limits). Trapezoidal Rule when the intervals are the same. You calculate h, then:. Trapezoidal Rule when the intervals are different. You calculate individual trapezoids and add them all up. LRAM - PowerPoint PPT Presentation
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Review 6
• Riemann Sums and Trapezoidal Rule
• Definite integrals (limits)
• Trapezoidal Rule when the intervals are the same.
• You calculate h, then:
hb b b b bn n2
2 2 21 2 3 1 . .
• Trapezoidal Rule when the intervals are different.
• You calculate individual trapezoids and add them all up.
LRAM
Calculate h, then add up all of the heights, starting on the far left and ending just prior to the endpoint.
• RRAM
Calculate h, then add up all of the heights, starting on the far right and ending just prior to the endpoint.
MRAM
Calculate h, then add up all of the heights by finding the midpoint of each individual interval.
• A test plane flies in a straight line with positive velocity v(t) in miles per minute. Selected values are given.
• All are counting by 5 minutes, so h = 5 mintues – even intervals.
t (min) 0 5 10 15 20 25 30 35 40
v(t) (mpm) 7 9.2 9.5 7 4.5 2.4 2.4 4.3 7.3
A. Give an estimate for the total miles traveled using a left sum.
• Always write out the integral.
• Total miles traveled in 40 minutes.
t (min) 0 5 10 15 20 25 30 35 40
v(t) (mpm) 7 9.2 9.5 7 4.5 2.4 2.4 4.3 7.3
v t d t v v v v v v v v( ) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )) . 0
40
5 0 5 10 15 20 25 30 35 231 5
B. Give an estimate for the total miles traveled using a right sum.
• Total miles traveled in 40 minutes.
t (min) 0 5 10 15 20 25 30 35 40
v(t) (mpm) 7 9.2 9.5 7 4.5 2.4 2.4 4.3 7.3
v t d t v v v v v v v v( ) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )) 5 5 10 15 20 25 30 35 40 2330
40
C. Give a total estimate using 4 sub intervals of equal length using a midpoint sum.
• Total miles traveled in 40 minutes.
t (min) 0 5 10 15 20 25 30 35 40
v(t) (mpm) 7 9.2 9.5 7 4.5 2.4 2.4 4.3 7.3
v t d t v v v v( ) ( ( ) ( ) ( ) ( )) 10 5 15 25 35 2290
40
D. Use a trapezoidal sum to calculate the total distance traveled.
• Total miles traveled in 40 minutes.
t (min) 0 5 10 15 20 25 30 35 40
v(t) (mpm) 7 9.2 9.5 7 4.5 2.4 2.4 4.3 7.3
v t d t v v v v v v v v v( ) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )) . 5
20 2 5 2 10 2 15 2 20 2 25 2 30 2 35 40 232 25
0
40
2008 #2 revisted.• Concert tickets are sold starting at noon (t = 0)
and are sold out in 9 hours. The number of people waiting in line can be modeled by the following table.
t (hours) 0 1 3 4 7 8 9
L(t) (people) 120 126 176 126 150 80 0
A. Find the total number of people waiting in line using a left sum.
• people
t (hours) 0 1 3 4 7 8 9
L(t) (people) 120 126 176 126 150 80 0
L t d t L L L L L L( ) ( ( )) ( ( )) ( ( )) ( ( )) ( ( )) ( ( )) 1 0 2 1 1 3 3 4 1 7 1 8 12160
9
B. Find the total number of people waiting in line using a right sum.
• people
t (hours) 0 1 3 4 7 8 9
L(t) (people) 120 126 176 126 150 80 0
L t d t L L L L L L( ) ( ( )) ( ( )) ( ( )) ( ( )) ( ( )) ( ( )) 1 9 1 8 3 7 1 4 2 3 1 1 11640
9
C. Find the total number of people waiting in line using a midpoint sum with three
intervals.• Midpoint – 3 intervals so the midpoints are
inbetween 0-3, 3-7, 7-9. The first midpoint will be 3 wide, then 4 wide, then 2 wide.
people
t (hours) 0 1 3 4 7 8 9
L(t) (people) 120 126 176 126 150 80 0
L t d t L L L( ) ( ( )) ( ( )) ( ( )) 3 1 4 4 2 8 11320
9
D. Use a trapezoidal sum to estimate the number of people waiting in line for the first
4 hours.
people
t (hours) 0 1 3 4 7 8 9
L(t) (people) 120 126 176 126 150 80 0
L t d t L L L L L L( ) ( ( ) ( )) ( ( ) ( )) ( ( ) ( )) 12
22
12
0
4
0 1 1 3 3 4 621
Average? 1/(b-a) times the answer• Find the average number of people waiting in line for the
first 4 hours using a trapezoidal sum.
• people
t (hours) 0 1 3 4 7 8 9
L(t) (people) 120 126 176 126 150 80 0
1
14 0
14
0
4
12
22
120 1 1 3 3 4 155 25
b a
a
b
f t d t
L t d t L L L L L L
( )
( ) ( ( ( ) ( )) ( ( ) ( )) ( ( ) ( ))) .
1
b af x dx
a
b
( )
Average Value of f(x) on [a, b]
f x dx F b F aa
b
( ) ( ) ( )
•Fundamental Theorem
of Calculus Part 2
Solving for area under the curve
• Find the anti-derivative• Evaluate with the limits• Units? The time goes away if it’s velocity, or the
power decreases on the time if it’s acceleration.• Units – unit’s squared – measurement squared.• Might be pulling information from a graph! It’s
the area under the curve!• If a starting point is given, then add up all of the
areas as is! + and – as given.
( )
( ) ( )( )
x dx
xx
2
1
2
3 1
2
23
13
4
4
4 2 4 1 15
3
3 3
• Fundamental Theorem of Calculus, Part 1
• means
g x f t d ta
x
( ) ( ) f t g x( ) ( )
Example a
• The graph of f is given. Let g be the function given by
• This means that f(t)=g’(x). So whenever you need a value for g’(x) you just read the graph. g(x) will be the area under the curve, starting at 2.
g x f t d tx
( ) ( ) 2
Example a (continued)
• Find g(3), g’(3) and g’’(3).• Write the tangent line for x = 3.
• g(3) is the integral, g’(3) is on the graph, g’’(3) is the slope of the curve that goes through 3.
Example a continued• So g(3)
• g’(3) = 2 (from the graph)• g’’(3) is
• The point is (3, 3) with a tangent slope of 2 so the tangent line is y = 2(x – 3) +3
0 44 2 2
g f t d t( ) ( ) ( )3 4 2 312
2
3
Example b• The graph of f consists of six line
segments. Let g be the function given by
• Again, f(t) = g’(x)• The function is starting at 0, so all area
calculations must start at 0.
g x f t d tx
( ) ( ) 0
Example b (continued)• Find g(4), g’(4), g’’(4)
• g(4) is the integral which will be made up of a triangle that has negative area and a trapezoid.
• g’(4) is from the graph and is 0
• g’’(4) is the slope of the curve going through 40 24 3 2
g f t d t( ) ( ) ( )( ) ( )4 1 2 3 1 312
22
0
4
Example C
• The graph of f consists of three line segments. Let g be the function given by
• And let h be the function given by
• Same pattern as previous 2 examples.
g x f t d tx
( ) ( ) 4
h x f t d tx
( ) ( ) 3
Example c (continued)
• Find g(1) and g’(1)
• g(1) is the integral
• g’ (1) is from the graph which is 2.
g f t d t( ) ( ) ( ) ( )( ) ( )( )1 4 2 2 1 2 1 932
12
12
4
1
• Find all intervals where h is decreasing.
• Since the integral has x in the lower limit, the function must be read right to left, or you make the integral negative and switch places with the limits. Thus, anytime the curve is above the x-axis, the actual function is decreasing. You must mention that h’ = -f whenever f >0,. So h decreases from (0, 2)
Related Documents
