Discuss how you would find the area under this curve!

APROXIMATE Area under a 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?

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

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

SOLUTION (1)(1)+(4)(1)+(9)(1)=14 Is this an overestimate or an underestimate? What if the function had been a decreasing function? Under

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 Solution Find 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. The middle x-values in each region

CAUTION!!!! MRAM ≠ (LRAM + RRAM)/2

Complete the following table: Increasing Function Decreasing LRAM RRAM

Under/Over Estimates for Increasing Function LRAM produces an _____________ underestimate RRAM produces an _____________ overestimate

Under/Over Estimates for Decreasing Function LRAM produces an _____________ overestimate RRAM produces an _____________ underestimate

How do we get the best underestimate of the area between a and 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 [0,2] using 8 partitions. What is the width of each partition? 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?

How could we make these approximations more accurate? Area Under Curves You Try: How could we make these approximations more accurate?

Let’s look at a HW problem . . . Unit 5 HW Document

LRAM - Using a table Partitions x 3 6 9 12 f(x) 20 30 25 40 37 3 6 9 12 f(x) 20 30 25 40 37 Partition [0,3] [3,6] [6,9] [9,12] Width of interval f(x) value at left end of interval.

RRAM - Using a table Partitions x 3 6 9 12 f(x) 20 30 25 40 37 3 6 9 12 f(x) 20 30 25 40 37 Partition [0,3] [3,6] [6,9] [9,12] Width of interval f(x) value at right end of interval.

Midpoint Rectangle Sums (MRAM) Use the middle function value in each interval. Ex: Calculate the Midpoint Sum using 3, equal-width partitions. Partitions x 2 4 6 8 10 12 f(x) 20 30 25 40 42 32 37 Partition midpoints

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 . . . .

Example Problem: 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. Time (seconds 2 4 6 8 10 Speed (ft/sec) 30 36 40 48 54 60 “never decreasing”— Scatterplot

Example Problem: 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. Time (sec) 2 4 6 8 10 Speed (ft/sec) 30 36 40 48 54 60 1. What is the best lower estimate for the distance the car traveled in the first 2 seconds? “lower estimate”--_________ since increasing function

Example Problem: 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. Time (sec) 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) 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) 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) 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) 2 4 6 8 10 Speed (ft/sec) 30 36 40 48 54 60

Riemann Sums These 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?