CALCULUS AB
DO NOW:
Sections 4.2 – 4.6 (day 2)
(Show all needed work in your notebook)
2
Refer to the region R enclosed between the graph of the function y = 2 x − x and the
x-axis for 0 ≤ x ≤ 2 .
1.
Partition [0, 2] into 4 subintervals and show the four rectangles that LRAM uses
to approximate the area of R.
Compute the LRAM sum without a calculator.
2.
Repeat Exercise 1 for RRAM and MRAM.
3.
Using a calculator program, find the RAM sums that complete the following table.
n
10
50
100
500
LRAM
MRAM
RRAM
The concept of: Riemann Sum
When we refer to the concept of Riemann Sum, the length of the n subintervals in
which we portioned the interval [a, b] do not have to be necessarily the same, as when
we use LRAM, RRAM, and MRAM (which are special cases of Riemann Sums).
When the number of subintervals n approaches infinite, the Riemann Sum will
approach the area under the curve, or the are between the curve and the x-axis,
independently on how the partition of the interval [a, b] is made.
(The intervals do not need to have equal length)
CLASSWORK:
1.
2
Approximate the area bounded by f ( x) = 4 − x and the x-axis using
a)
LRAM and n = 4.
b)
RRAM and n = 4.
c)
MRAM and n = 4.
2.
The table shows the velocity of a remote-controlled toy car as it traveled down a
hallway for 10 seconds.
Time (sec)
Velocity (in./sec)
0
0
1
6
2
10
3
16
4
14
5
12
6
18
7
22
8
12
9
4
10
2
Estimate the distance traveled by the car using 10 subintervals of length 1 and
the methods shown.
a)
b)
LRAM
RRAM
3.
The table shows the rate in liters/min at which water leaked out of a container.
0
5.6
Time (min)
Rate (liters/min)
1.2
4.3
2.3
3.1
3.8
2.2
5.4
1.5
Aright-hand Riemann sum is computed using the four subintervals indicated in
the table. This Riemann sum estimates the total amount of water that has leaked
out of the container. What is the estimate?
A)
D)
12.70 liters
16.95 liters
B)
E)
14.27 liters
19.62 liters
C)
16.70 liters
4.
The temperature, in degrees Celsius (˚C), of a turkey in an oven is a continuous
function of time t. Some values of this function are given in the table.
Time (min)
Temperature (˚C)
0
24
5
76
10
106
15
124
20
135
Approximate the average temperature (in ˚C) of the turkey over the interval
0 ≤ t ≤ 25 using a LHRS with subintervals of length 5 minutes.
25
141