Section 5.1: Estimating with Finite Sums Objectives: Students will be able to… Find distance traveled Estimate using Rectangular Approximation Method Estimate the Volume of a sphere

Distance Traveled If you were traveling at a constant rate of 60 mph for 3 hours, how far did you travel? Let’s look at a graph, shall we???

Rectangular Approximation Method (RAM) Can find total area under the curve by dividing up the interval into subintervals (little rectangles) and adding up the area of each subinterval (rectangle) If you have a function over interval [a,b]: Width of rectangle= , where n is the number of subintervals. Height of rectangle = the value of the function at either the right endpoint, left endpoint, or midpoint of the interval As n →∞, the more accurate your approximation

Rectangular Approximation Method LRAM: left endpoint RAM RRAM: right endpoint RAM MRAM: midpoint RAM

The rate of sales (in games per week) of a new video game is shown in the table below. Assuming that the rate of sales increased throughout the 20-week period, estimate the total number of games sold during this period. Time weeks 5 10 15 20 Rate of sales games/week 585 892 1875 2350

A car is moving with increasing velocity and we measure the car’s velocity every two seconds. How far has the car traveled? Time (sec) 2 4 6 8 10 Velocity (ft/sec) 20 30 38 44 48 50

Example: The velocity in m/s of an object moving along a straight line is given by the function v=t2, where 0 < t < 8. Approximate the displacement of the object by dividing the interval [0, 8] into n subintervals of equal length. Use n= 8, and MRAM.

Look at the graph of the function y = -x2 +4 on [0, 2]. Divide into 8 subintervals Is the function inc or dec over interval? Will RRAM be overestimate or underestimate? LRAM? Find the area under the curve using RRAM.

General guidelines INCREASING FUNCTION: RRAM overestimates, LRAM underestimates DECREASING FUNCTION: RRAM underestimates, LRAM overestimates.

Use LRAM, RRAM, and MRAM to evaluate the area under the curve Use LRAM, RRAM, and MRAM to evaluate the area under the curve. A graph may help. y = x3 -5x2 +6x +5 on [0,4], use n=5.

Finding Volume of a Sphere Estimate the volume of a sphere with a radius of 4. Picture a semicircle with a radius of 4 revolving around the x-axis. What does each subinterval create when revolved?