Ch. 6 – The Definite Integral

6.1 – Estimating with Finite Sums

• Ex: A car is traveling at a speed of 20 m/s from time t = 2 s to time t = 4 s. How much distance does the car travel from t = 2 to t = 4? – When velocity is constant, this is easy…use v = d/t to get d =

40 m.– How can we use a graph of velocity vs. time to find the answer

to the example above?• Since v times t is distance, find the area of the rectangle

under the line over the correct interval!

– Secondary question: in terms of calculus, how are d(t) and v(t) related?

• v is the derivative of d!

– Using the example at right, how dowe find the antiderivative of a function using its graph?

• Find the area under the curve! 4

v(t) (m/s)

t (s)2

20

40 m

2 s

20 m/s

• Given a graph of f(x),– Find the derivative of f by calculating the slope of f at

every point (x, f(x)).– Find the antiderivative of f by calculating the area

between f and the x-axis between 0 and x.

• If f is not a straight line, this can be difficult because no geometric area formula may be useful:

• To find this area, we will add together the areas of a whole bunch of skinny rectangles!

f(x)

x1xx2

• LRAM = Left-Hand Rectangular Approximation Method

• RRAM = Right-Hand R.A.M.

• MRAM = Midpoint R.A.M.

Height of rectangles intersect curve in the

top left corner

Height of rectangles intersect curve in the

top right corner

Height of rectangles intersect curve in the midpoint of the top

Width of rectangles is the same for

each rectangle!!!!

• Ex : Estimate the area of the region bounded by the graph of f(x) = 9 – x2 and the x- and y-axes using LRAM and 6 subintervals. – Each width is 3/6 = 0.5– How do we find the heights of the rectangles?– Use f(0), f(0.5), f(1), …, f(2.5)

for the height of the rectangles…– LRAM = (0.5)(9) + (0.5)(8.75) + …

+ (0.5)(5) + (0.5)(2.75)• Don’t use f(3) because it’s not a left-hand value for any of the rectangles

– LRAM = (0.5) (9 + 8.75 + … + 5 + 2.75)

– LRAM = 20.125

                                                                                          

x f(x)0 90.5 8.751 8

1.5 6.752 52.5 2.75

3 0

x = 3

• Ex (cont.): Estimate the area of the region bounded by the graph of f(x) = 9 – x2 and the x- and y-axes using RRAM and 6 subintervals. – Each width is 3/6 = 0.5– How do we find the heights of the rectangles?– Use f(0.5), f(1), …, f(2.5), f(3)

for the height of the rectangles…– RRAM = (0.5)(8.75) + (0.5)(8) + …

+ (0.5)(2.75) + (0.5)(0)• Don’t use f(0) because it’s not a right-hand value for any of the rectangles

– RRAM = (0.5) (8.75 + 8 + … + 2.75+ 0)

– RRAM = 15.625

– For MRAM, you’d use f(0.25),f(0.75), …, f(2.25), f(2.75) for the heights

                                                                                          

x f(x)0 90.5 8.751 8

1.5 6.752 52.5 2.75

3 0

x = 3

• Ex: Use a calculator to estimate the area of the region bounded by the graph of f(x) = 2 + xsinx over [0, 7π/6] using the following RAM and subintervals:

– Actual area is about 10.004532 square units– Notice that as n increases, the RAMs become almost identical,

and they will be increasingly accurate.

n (# subintervals) LRAMn MRAMn RRAMn

2

10

100

1000

10.57435 10.59039 7.2159510.29912 10.02520 9.62744

10.00788 10.00453 10.0011710.03770 10.00473 9.97053