Sr. lecturer in mathematics KHEMUNDI COLLEGE , DIGAPAHANDI. Riemann Sums and the Definite Integral Prepared By Sri. Santosh Kumar Rath. Sr. lecturer in mathematics KHEMUNDI COLLEGE , DIGAPAHANDI.
Review We partition the interval into n sub-intervals Evaluate f(x) at right endpoints of kth sub-interval for k = 1, 2, 3, … n f(x)
Review a b 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 subintervals a = 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|| Choose an arbitrary value from each subinterval, call it
Riemann Sum 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
The Riemann Sum Calculated Consider the function 2x2 – 7x + 5 Use x = 0.1 Let the = left edge of each subinterval Note the sum
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
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 Subdivision rule f(x) a c b
Area As An Integral The area under the curve on the interval [a,b] A f(x) 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