Presentation on theme: "Riemann sums, the definite integral, integral as area"— Presentation transcript:
1 Riemann sums, the definite integral, integral as area Section 5.2a
2 First, we need a reminder of sigma notation: How do we evaluate:…and what happens if an “infinity” symbol appearsabove the sigma??? The terms go on indefinitely!!!LRAM, MRAM, and RRAM are all examples ofRiemann sums, because of how they were constructed.In this section, we start with a more general accountof these sums………………….observe…………….
3 We start with an arbitrary function f(x), defined on a closed interval [a, b].Partition the interval [a, b] into n subintervals by choosing n – 1points between a and b, subject only toabLetting a = x and b = x , we have apartition of [a, b]:n
4 We start with an arbitrary function f(x), defined on a closed interval [a, b].The partition P determines n closed subintervals.The subinterval is , which has lengthxIn each subinterval we choose some number, denoting thenumber chosen from the subinterval byOn each subinterval, we create a rectangle that reaches fromthe x-axis to touch the curve at
5 On each subinterval, we create a rectangle that reaches from the x-axis to touch the curve atabOn each subinterval, we form theproduct(which can be positive,negative, or zero…)Area of each rectangle!!!
6 Riemann sum for f on the interval [a, b] Finally, take the sum of these products:This sum is called theRiemann sum for f on the interval [a, b]
7 Riemann Sums As with LRAM, MRAM, and RRAM, all Riemann sums for a given interval [a, b] will converge to common value, as long asthe subinterval lengths all tend to zero.To ensure this last condition, we require that the longestsubinterval (called the norm of the partition, denoted ||P||)tends to zero…
8 Definition: The Definite Integral as a Limit of Riemann Sums Let f be a function defined on a closed interval [a, b]. For anypartition P of [a, b], let numbers be chosen arbitrarily in thesubintervalsIf there exists a number I such thatno matter how P and the ‘s are chosen, then f is integrableon [a, b] and I is the definite integral of f over [a, b].
9 Theorem: The Existence of Definite Integrals In particular, if f is continuous, then choices about partitions and‘s don’t matter, as long as the longest subinterval tends tozero:All continuous functions are integrable. That is, if afunction f is continuous on an interval [a, b], then itsdefinite integral over [a, b] exists.This theorem allows for a simpler definition of the definite integralfor continuous functions. We need only consider the limit ofregular partitions (in which all subintervals have the samelength)…
10 The Definite Integral of a Continuous Function on [a, b] Let f be continuous on [a, b], and let [a, b] be partitioned into nsubintervals of equal length Then thedefinite integral of f over [a, b] is given bywhere each is chosen arbitrarily in the subinterval.
11 Integral Notation The Greek “S” is changed to an elongated Roman “S,” so that the integral retains its identity as a “sum.”This is read as “the integral from a to b of f of x dee x”or “the integral from a to b of f of x with respect to x”
12 Integral Notation The function is Upper limit the integrand of integrationx is the variableof integration (alsocalled a dummyvariable)IntegralSignLower limitof integrationWhen you find the valueof the integral, you haveevaluated the integralIntegral of ffrom a to b
13 A Quick Practice Problem The interval [–1, 3] is partitioned into n subintervals of equallength Let denote the midpoint of thesubinterval. Express the given limit as an integral.The function being integrated isover the interval [–1, 3]...
14 Definition: Area Under a Curve (as a Definite Integral If is nonnegative and integrable overa closed interval [a, b], then the area underthe curve of from a to b is theintegral of from a to b,
15 Practice Problem Evaluate the integral What is the graph of the integrand???From Geometry-Land:(0, 2)Area =(–2, 0)(2, 0)
16 What happens when the curve is below the x-axis? The area is negative!!!Area =when
17 For any integrable function, If an integrable function y = f (x) has both positive and negativevalues on the interval [a, b], add the areas of the rectanglesabove the x-axis, and subtract those below the x-axis:For any integrable function,= (area above x-axis) – (area below x-axis)
18 What happens with constant functions? If f (x) = c, where c is a constant, on theinterval [a, b], thenDoes this make sense graphically???Quick Example:
19 Practice Problems Use the graph of the integrand and areas to evaluate the given integral.
20 Practice Problems Use the graph of the integrand and areas to evaluate the given integral.
21 Practice Problems Use the graph of the integrand and areas to evaluate the given integral.3b3aab