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Riemann Sums and the Definite Integral

- Lesson 5.3

Review

- We partition the interval into n sub-intervals
- Evaluate f(x) at right endpointsof kth

sub-interval for k 1, 2, 3, n

f(x)

Review

- Sum
- We expect Sn to improve thus we define A, the

area under the curve, to equal the above limit.

f(x)

Riemann Sum

- Partition the interval a,b into n

subintervalsa x0 lt x1 lt xn-1lt xn b - Call this partition P
- The kth subinterval is ?xk xk-1 xk
- Largest ?xk is called the norm, called P
- Choose an arbitrary value from each

subinterval, call it

Riemann Sum

- Form the sumThis 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

The Riemann SumCalculated

- Consider the function2x2 7x 5
- Use ?x 0.1
- Let the left edgeof each subinterval
- Note the sum

The Riemann Sum

f(x) 2x2 7x 5

- We have summed a series of boxes
- If the ?x were smaller, we would have gotten a

better approximation

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

Example

- Try
- Use summation on calculator.

Example

- Note increased accuracy with smaller ?x

Limit of the Riemann Sum

- The definite integral is the

limit of the Riemann sum.

Properties of Definite Integral

- Integral of a sum sum of integrals
- Factor out a constant
- Dominance

Properties of Definite Integral

f(x)

- Subdivision rule

a

c

b

Area As An Integral

f(x)

- The area under the curve on theinterval a,b

A

a

c

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

Assignment

- Section 5.3
- Page 348
- Problems 1, 5, 9, 11, 13, 17, 21, 25

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