5.1 Approximating and Computing Area
Thurs Jan 29
Evaluate each summation1)
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 for on the interval [1,3]
Ex
• Calculate for the same function [1,3]
Ex
• Calculate for on [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 #5 11 15 30 31 43 47 86