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