riemann sums and the definite integral lesson 5.3
TRANSCRIPT
Review
• We partition the interval into n sub-intervals
• Evaluate f(x) at right endpointsof kth sub-interval for k = 1, 2, 3, … n
a b
f(x)
b ax
n
a k x
Review
• Sum
• We expect Sn to improve thus we define A, the area under the curve, to equal the above limit.
a b
1
lim ( )n
nn
k
S f a k x x
f(x)
Look at Goegebra demo
Look at Goegebra demo
Riemann Sum
1. Partition the interval [a,b] into n subintervalsa = x0 < x1 … < xn-1< xn = b
• Call this partition P
• The kth subinterval is xk = xk-1 – xk
• Largest xk is called the norm, called ||P||
2. Choose an arbitrary value from each
subinterval, call it ic
Riemann Sum3. Form the sum
This is the Riemann sum associated with• the function f• the given partition P• the chosen subinterval representatives
• We will express a variety of quantities in terms of the Riemann sum
1 1 2 21
( ) ( ) ... ( ) ( )n
n n n i ii
R f c x f c x f c x f c x
1 1 2 21
( ) ( ) ... ( ) ( )n
n n n i ii
R f c x f c x f c x f c x
ic
The Riemann SumCalculated
• Consider the function2x2 – 7x + 5
• Use x = 0.1
• Let the = left edgeof each subinterval
• Note the sum
x 2x 2̂-7x+5 dx * f(x)4 9 0.9
4.1 9.92 0.9924.2 10.88 1.0884.3 11.88 1.1884.4 12.92 1.2924.5 14 1.44.6 15.12 1.5124.7 16.28 1.6284.8 17.48 1.7484.9 18.72 1.872
5 20 25.1 21.32 2.1325.2 22.68 2.2685.3 24.08 2.4085.4 25.52 2.5525.5 27 2.75.6 28.52 2.8525.7 30.08 3.0085.8 31.68 3.1685.9 33.32 3.332
Riemann sum = 40.04
x 2x 2̂-7x+5 dx * f(x)4 9 0.9
4.1 9.92 0.9924.2 10.88 1.0884.3 11.88 1.1884.4 12.92 1.2924.5 14 1.44.6 15.12 1.5124.7 16.28 1.6284.8 17.48 1.7484.9 18.72 1.872
5 20 25.1 21.32 2.1325.2 22.68 2.2685.3 24.08 2.4085.4 25.52 2.5525.5 27 2.75.6 28.52 2.8525.7 30.08 3.0085.8 31.68 3.1685.9 33.32 3.332
Riemann sum = 40.04
ic
The Riemann Sum
• We have summed a series of boxes
• If the x were smaller, we would have gotten a better approximation
f(x) = 2x2 – 7x + 5
1
( ) 40.04n
i ii
f c x
The Definite Integral
• The definite integral is the limit of the Riemann sum
• We say that f is integrable when the number I can be approximated as accurate
as needed by making ||P|| sufficiently small f must exist on [a,b] and the Riemann sum
must exist
0
1
lim( )b
a P
n
i ik
f f c xI x dx
Properties of Definite Integral
• Integral of a sum = sum of integrals
• Factor out a constant
• Dominance
( ) ( ) [ , ]
( ) ( )b b
a a
f x g x on a b
f x dx g x dx
Properties of Definite Integral
• Subdivision rule
( ) ( ) ( )c b c
a a b
f x dx f x dx f x dx
a b c
f(x)
Distance As An Integral
• Given that v(t) = the velocity function with respect to time:
• Then Distance traveled can be determined by a definite integral
• Think of a summation for many small time slices of distance
( )t b
t a
D v t dt