Trapezoidal Approximation Calculus AP Unit 5 Day 3 Trapezoidal Approximation

Draw this sketch in your notes NOTE: PPT slides for posting purposes. Found document camera to work the best for lesson

Let’s examine this picture . . . The curve is the blue wave. There are four trapezoids drawn. Label the bases and “height” of each trapezoid. Determine the area of each trapezoid using . The sum of these trapezoidal areas approximates the area between the curve and the x-axis. Write this sum. Simplify this sum expression. NOTE: PPT slides for posting purposes. Found document camera to work the best for lesson

“Trapezoidal Rule” Summary for 4 partitions b1 b4 b2 b3 b0 h h h h

Function Behavior and Over/Under Trapezoidal Approximation Recall, that LRAM and RRAM approximations are over/under based on the increasing/decreasing behavior of the function. Examine the concavity of the above sketch to make a statement about when a trapezoidal approximation is an over estimate. Make a statement about when a trapezoidal approximation is an under estimate.

Function Behavior and Over/Under Trapezoidal Approximation When the graph of the function is concave down ( ), the trapezoidal approximation is an under estimate. When the graph of the function is concave up ( ), the trapezoidal approximation is an over estimate.

Example Problem: (#1-2) Trapezoidal Approximations 1. Using the function from [0,3] to the right, finish drawing in trapezoids by connecting the endpoints of the partitions. Assume there are three partitions. 2. Looking at these, do you think adding up the trapezoids gives a more accurate or less accurate area estimate than the rectangles did yesterday? WHY?

Example Problem: (#3-4) Trapezoidal Approximations 3. Write the formula for area of a trapezoid.   4. Use this formula to estimate the area of the three trapezoids in the above drawing. Add them up to get an estimate for the area under this curve. Is this estimate an under estimate or over estimate of the actual area between the curve and the x-axis? Justify your answer.

Example Problem: (#5) Trapezoidal Approximations Use n=3 trapezoids to approximate the area between the graph of and the x-axis on the interval [0,9]

Example Problem: (#6) Trapezoidal Approximations 6. . . . . . Try to write a general equation for the area under a curve using trapezoids. Use b0 for the first base, b1 for the second base, b3 for the third base, ……up to bn for the nth base. “n” partitions

Trapezoidal Rule for “n” partitions

Trapezoidal Rule—Example 8. Coal gas is produced at gasworks. Pollutants in the gas are removed by scrubbers, which become less and less efficient as time goes on. The following measurements, made at the start of each month, show the rate at which pollutants are escaping (in tons/month) in the gas: Use trapezoidal rule to estimate the quantity of pollutants that escaped during the first three months. During all six months.

The Link to LRAM and RRAM DON’T make the common mistake!!!

Notes The EXACT area between a curve, y=f(x), and x-axis from a to b is the INTEGRAL of f from a to b: Means find the area between the x-axis and y = x4 over the interval [2,6] Example:

Important If the value of an integral is negative the area would be below the x-axis. In this case, the actual area would be the absolute value of each integral. We will use the calculator to explore this idea.

fnInt on the Calculator MATH #9 OR on NEW Operating System, HOT button at the top of the calculator key pad ALPHA F2 Function to integrate Lower bound fnInt(y1, x,a, b) Upper bound Integration variable

fnInt on the Calculator Function to integrate Lower bound fnInt(y1, x, 0, 6) Upper bound Integration variable

fnInt on the Calculator Function to integrate Lower bound fnInt(y1, x, -2, 0) Upper bound Integration variable

BUT the area between f(x)=x3 and the x-axis is: