5.1 Approximating and Computing Area Thurs Jan 29 Evaluate each summation 1) 2)

Distance and velocity Distance traveled = velocity x time elapsed If velocity changes (is a function): Then distance traveled = f(t) x dt or the area under the graph of f(t) over [t1, t2] Note: in this scenario we are going from a rate to not a rate

Area under a curve So finding the area under a curve allows us to calculate how much has accumulated over time (common AP free response) First we will discuss ways to approximate this accumulation

Approximating Area Although the area under a graph is usually curved, we can use rectangles to approximate the area. More rectangles = better approximation

3 types of rectangle approx. The 3 approximations only differ in where the height of each rectangle is determined Left-endpoint: use the left endpoint of each rectangle to determine height Right-endpoint: use the right endpoint of each rectangle Midpoint: use the midpoint of the endpoints

Rectangle Approx. 1) Determine and each interval 2) Use the correct endpoint to find the height of each rectangle 3) Add all the heights together 4) Multiply sum by the width ( )

endpoint formulas The formula for the Nth right-endpoint approximation: The formula for the Nth left-endpoint approx:

Ex Calculate foron the interval [1,3]

Ex Calculate for the same function [1,3]

Ex Calculate foron [2,4]

Interpretations Which approximation is the best? – Depends on the function – Left and Right endpoints can overestimate or underestimate depending on if the function increases or decreases – Midpoint is typically the safest because it does both so it averages out

Summation Review Summation notation is standard for writing sums in compact form

Approximations as Sums As sums,

Summation Theorems If n is any positive integer and c is any constant, then: Sum of constants Sum of n integers Sum of n squares

Summation Theorems For any constants c and d,

Computing Actual Area So far all we’ve been doing is adding areas of rectangles, where does the calculus come in? As the # of rectangles approach infinity, we can find the actual area under a curve

Theorem – not covered on AP If f(x) is continuous on [a, b], then the endpoint and midpoint approximations approach one and the same limit as There is a value L such that

Ex 1 Find the area under the graph of f(x) = x over [0,4] using the limit of right-endpoint approx.

Ex 2 Let A be the area under the graph of f(x) = 2x^2 – x + 3 over [2,4]. Compute A as the limit

Closure Approximate the area under f(x) = x^2 under the interval [2, 4] using left-endpoint approximation and 4 rectangles HW: p.296 #