Discuss how you would find the area under this curve!

Discuss how you would find the area under this curve!

AP Calculus Unit 5 Day 2

APROXIMATE Area under a Curve

• We will introduce some new Calculus concepts–LRAM–RRAM–MRAM

Rectangular Approximation Method (RAM) to estimate area under curve

Example Problem: Use LRAM with partition widths of 1 to estimate the area of the region enclosed between the graph of and the x-axis on the interval [0,3].

NOTICE—This is an underestimate. What if the function had been a decreasing function?

2( )f x x

Rectangular Approximation Method (RAM) to estimate area under curve SOLUTION

Example Problem: Use LRAM with partition widths of 1 to estimate the area of the region enclosed between the graph of and the x-axis on the interval [0,3].

NOTICE—This is an underestimate. What if the function had been a decreasing function? over

2( )f x x

(0)(1)+(1)(1)+(4)(1)=5

Is this an overestimate or an underestimate? What if the function had been a decreasing function?

Is this an overestimate or an underestimate? What if the function had been a decreasing function? Under

SOLUTION

(1)(1)+(4)(1)+(9)(1)=14

BEWARE—MRAM uses the average of the two x-values NOT the two y-values!

MRAM gives the best estimate regardless of whether the function is increasing or decreasing.

Midpoint Method SolutionFind the area between the x-axis and y = x2 between [0,3] with partition widths of 1.

MRAM: The Midpoint of each rectangle touches the graph.

x 1.5

f 1.5 2 2 21 0.5 1 1.5 1 2.5 8.75

The middle x-values in each region

CAUTION!!!!MRAM ≠ (LRAM + RRAM)/2

Complete the following table:Increasing Function

Decreasing Function

LRAM

RRAM

Under/Over Estimates for Increasing Function

LRAM produces an _____________underestimate RRAM produces an _____________overestimate

LRAM produces an _____________overestimate RRAM produces an _____________underestimate

Under/Over Estimates for Decreasing Function

How do we get the best underestimate of the area between a and b?

a b

Answer: Divide into 2 intervals (increasing/decreasing) and pick LRAM or RRAM for each part that yields an underestimate for that half.

Together let’s practice with finding partition widths

Find the upper and lower approximations of the area under

on the interval [1,2] using 5 partitions. What is the width of each partition?

( ) 2f x x Find the upper and lower approximations of the area under on the interval [0,2] using 8 partitions. What is the width of each partition?

1( )f xx

Area Under Curves

How could we make these approximations more accurate?

You Try:

3 0 3 3 3 6 12.545

3 3 3 6 3 9 21.545

3 1.5 3 4.5 3 7.5 18.254

LRAM

RRAM

MRAM

Let’s look at a HW problem . . .

Unit 5 HW Document

Problem #5

LRAM - Using a table

x 0 3 6 9 12f(x) 20 30 25 40 37

LRAM 3 20

Partitions

Partition [0,3] [3,6] [6,9] [9,12]

3 30 3 25 3 40 345

f(x) value at left end of interval.

Width of interval

RRAM - Using a table

x 0 3 6 9 12f(x) 20 30 25 40 37

RRAM 3 30

Partitions

Partition [0,3] [3,6] [6,9] [9,12]

3 25 3 40 3 37 396

f(x) value at right end of interval.

Width of interval

Midpoint Rectangle Sums (MRAM)

x 0 2 4 6 8 10 12f(x) 20 30 25 40 42 32 37

Partitions

Partition midpoints

MRAM 4 30 4 40 4 32 408

Use the middle function value in each interval.

Ex: Calculate the Midpoint Sum using 3, equal-width partitions.

NOTES: Background Problem

• A car is traveling at a constant rate of 55 miles per hour from 2pm to 5pm. How far did the car travel?– Numerical Solution—

– Graphical Solution—

• But of course a car does not usually travel at a constant rate . . . .

A car is traveling so that its speed is never decreasing during a 10-second interval. The speed at various points in time is listed in the table below.

Example Problem:

Time (secon

ds

0 2 4 6 8 10

Speed (ft/sec

)

30 36 40 48 54 60

“never decreasing”—

Scatterplot

A car is traveling so that its speed is never decreasing during a 10-second interval. The speed at various points in time is listed in the table below.

Example Problem:

1. What is the best lower estimate for the distance the car traveled in the first 2 seconds?

“lower estimate”--_________ since increasing function

Time (sec)

0 2 4 6 8 10

Speed (ft/sec)

30 36 40 48 54 60

A car is traveling so that its speed is never decreasing during a 10-second interval. The speed at various points in time is listed in the table below.

Example Problem:

Time (sec)

0 2 4 6 8 10

Speed (ft/sec)

30 36 40 48 54 60

2. What is the best upper estimate for the distance the car traveled in the first 2 seconds?

You Try: 3. What is the best lower estimate for the total distance traveled during the first 4 seconds? (Assume 2 second intervals since data is in 2 second intervals) 

Time (sec)

0 2 4 6 8 10

Speed (ft/sec)

30 36 40 48 54 60

You Try: 4. What is the best upper estimate for the total distance traveled during the first 4 seconds? 

Time (sec)

0 2 4 6 8 10

Speed (ft/sec)

30 36 40 48 54 60

You Try: 5. Continuing this process, what is the best lower estimate for the total distance traveled in the first 10 seconds? 

Time (sec)

0 2 4 6 8 10

Speed (ft/sec)

30 36 40 48 54 60

You Try: 6. What is the best upper estimate for the total distance traveled in the first 10 seconds?

Time (sec)

0 2 4 6 8 10

Speed (ft/sec)

30 36 40 48 54 60

Riemann SumsThese estimates are sums of products and are known as Riemann Sums. 7. If you choose the lower estimate as your approximation of how far the car traveled, what is the maximum amount your approximation could differ from the actual distance?