5.1 Estimating with Finite Sums Objectives SWBAT: 1) approximate the area under the graph of a nonnegative continuous function by using rectangular approximation methods 2) interpret the area under a graph as a net accumulation of a rate of change

All year long we have been finding derivatives. We will now begin undoing them by finding definite integrals. To lay the foundation, we are going to discuss Finite Sums (5.1) and Riemann Sums (5.2).

Example 1: Suppose Carucci is LTD, and is taking a road trip to the Seven Layers of the Candycane Forest. From the 2 nd to the 4 th hour of his road trip, he travels with the cruise control in his Fit set to exactly 70 miles per hour. How far did he travel during this time? (Hint: you do not need Calculus to figure this out) Example 2: Sketch a graph modeling the situation above. Geometrically, how can we indicate the total distance traveled?

What happens when velocity is not constant? The area is no longer a rectangle, as it was in the previous example. Our technique here will be similar in that we will break the time intervals into much shorter segments so that the velocity over those time segments is almost constant. We can then find the distance traveled for each time interval (the area of a thin rectangle) and add all the areas of the rectangles together. This will give us the area under the curve.

There are several ways for us to approximate the area under a curve. Section 5.1 deals with: – LRAM (Left-hand Rectangular Approximation Method) – RRAM (Right-hand Rectangular Approximation Method) – MRAM (Midpoint Rectangular Approximation Method) Section 5.5 deals with the Trapezoid Rule.

The Area Problem and the Rectangular Approximation Method (RAM) (aka Riemann Sums) Step 1: Divide the interval from a to b into subintervals (the number of which is arbitrary) Step 2: Draw rectangles under the curve. How you draw the rectangles will be up to you or it will be dictated to you. Step 3: Find the areas of the rectangles. The width is the specified interval width, and the height is the function value at the x value on the interval. Step 4: Add the areas. This will give us the approximate area under the curve. The sums are known as Riemann Sums

Example 4: What method is shown in the graphs below? The left-hand side of each interval is used to measure the height of each rectangle. So this uses LRAM. Also uses LRAM. Follow up question! Which graph gives a better approximation of the area under the curve? The graph on the right!!! More rectangles means better approximation.

Example 5: Illustrate the use of RRAM and MRAM on the graphs below. Use 4 rectangles.

Example 7: The table below shows the velocity of a model train engine moving along a track for 10 seconds. a) Using a left Riemann Sum with 10 subintervals, estimate the distance traveled by the engine in the first 10 seconds. b) Using a Midpoint Riemann Sum with 5 subintervals, estimate the distance traveled by the engine in the first 10 seconds.