Riemann Sums and The Definite Integral 1

Area under a curve The other day we found the area under a curve by dividing that area up into equal intervals (rectangles of equal width). This section goes one step further and says we can divide the area into uneven intervals, and the same concept will still apply. It will also examine functions that are continuous, but are no longer entirely non-negative. 2

Let’s quickly review what we learned last time … 3

If f(x) is a nonnegative, continuous function on the closed interval [a, b], then the area of the region under the graph of f(x) is given by This is our limiting process where we let n approach infinity 4 4 4

Notice that through the use of rectangles of equal width, we were able to estimate the area under a curve. This curve was always non-negative (i.e. it was always above the x-axis); therefore, the values we got when we multiplied f(xi) by Δx were always positive. 5

Then take the sum of these products. The graph of a typical continuous function y = ƒ(x) over [a, b]: Partition [a, b] into n subintervals a < x1 < x2 <…xn < b. Select any number in each subinterval ck. Form the product f(ck)xk. Then take the sum of these products. 6

The sum of these rectangular areas is called the Riemann Sum of the partition of x. LRAM, MRAM, and RRAM are all examples of Riemann Sums. The width of the largest subinterval of a partition  is the norm of the partition, written ||x||. As the number of partitions, n, gets larger and larger, the norm gets smaller and smaller. As n, ||x|| 0 7

Formal Definition from p. 266 of your textbook 8

Finer partitions of [a, b] create more rectangles with shorter bases. 9

This limit of the Riemann sum is also known as the definite integral of f(x) on [a, b] This is read “the integral from a to b of f of x dx,” or “the integral from a to b of f of x with respect to x.” 10

variable of integration (differential) Notation for the definite integral upper limit of integration Integration Symbol (integral) integrand variable of integration (differential) lower limit of integration 11

Formal Definition from p. 267 of your textbook 12

NOW … Before we go any further, let’s make one quick – yet important – clarification regarding INDEFINITE INTEGRALS (with which we have worked before in finding antiderivatives) and DEFINITE INTEGRALS (which we are about to study now): 13

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Today we will focus solely on the DEFINITE integral. The connection between the two types of integrals will probably not be clear now, but be patient. We will explore and examine the connection between the two next time.  Woo-hoo! 15

Evaluate the following definite integrals using geometric area formulas. 16 16

Top half only! 17 17

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THEOREM: If f(x) is continuous and non-negative on [a, b], then the definite integral represents the area of the region under the curve and above the x-axis between the vertical lines x = a and x = b. a b 19

Rules for definite integrals 20

The Integral of a Constant 21

Rules for definite integrals “Preservation of Inequality” Theorem If f is integrable and non-negative on [a, b] then If f and g are integrable and non-negative on [a, b] and f(x) < g(x) for every x in [a,b], then 22

Using rules for definite integrals Example: Evaluate the using the following values: = 60 + 2(2) = 64 23

Using the TI 83/84 to check your answers Find the area under on [1,5] Graph f(x) Press 2nd CALC 7 Enter lower limit 1 Press ENTER Enter upper limit 5 Press ENTER. 24

When functions are non-negative, the Riemann sums represent the areas under the curves, above the x-axis, over some interval [a, b]. When functions are negative, however, the Riemann sums represent the negative (or opposite) values of those areas. In other words, Riemann sums DO have direction and CAN take on negative values. 25

To summarize that thought … f A a b A1 f A3 = area above – area below a b A2 26

ASSIGNMENT: p. 272 – 273 (14-22even, 24-30even, 31, 34-40even, 45) 27