
In the previous section, we saw that if f is continuous on an interval [a,b], then the endpoint and midpoint approximations approach a common limit L as N → ∞:
L = lim_{N → ∞} R_{N} = lim_{N → ∞ }L_{N} = lim_{N → ∞} M_{N}

When f(x) ≥ 0, L is the _____ under the graph of f. In a moment, we will state formally that L is the _____ _______ of f over [a,b]. Before doing so, we introduce more general approximations called ______ _____
 area
 definite integral
 Riemann sums

Recall that R_{N}, L_{N} and M_{N} use rectangles of equal width Δx, whose heights are the values of ____ at the endpoints or midpoints of the _______. In Riemann sum approximations, we relax these requirements: The rectangles need not have _____ width and height may be any value of ____ within the subinterval
 f(x)
 subintervals
 equal
 f(x)

To specify a Riemann sum, we choose a partion and a set of sample points (Define both)
 Partion: P of size N, a choice of points that divides [a,b] into N subintervals
 P: a = x0 < x1 < x2 <...<x_{N} = b
 Sample points: C = {c_{1},..., c_{N}}: c_{i} belongs to the subinterval [x_{i1}, x_{i}] for all i = 1,...,N

What is the length of i^{th} subinterval [x_{i1}, x_{i}]?
Δx_{i} = x_{i}  x_{i1}

Define the norm of P
Norm: denoted P, is the maximum of the lengths Δx_{i}

Given P and C, we construct the rectangle height ____ and base _____ over each subinterval [xi1, xi], as in Fig 2B (pg 38) This rectangle has area _______ if f(c) ≥ 0. If f(ci) < 0, the rectangle extends below the xaxis, and f(ci)Δxi is the _______ of its area
 f(ci)
 Δxi
 f(ci)Δxi
 negative

Represent the Riemann sum in summation notation
R(f,P,C) = ^{N}Σ_{i=1} f(c_{i})Δx_{i} = f(c_{1})Δx_{1} + f(c_{2})Δx_{2}+...+ f(c_{N})Δx_{N}

Keep in mind that RN, LN, and MN are particular examples of _______ _____ in which Δx_{i} = ______ for all i, and the sample points c_{i} are either _______ or _______
 Riemann sums
 (b  a)/N
 endpoints or midpoints

Note in Fig 2C (pg 238) that as the norm P tends to zero (meaning that the rectangles get ______), the number of rectangles N tends to _____ and they approximate the area under the graph more closely. Define integrable
 thinner
 ∞
 Integrable: f is integrable over [a,b] if all the Riemann sums (not just the endpoint and midpoint approximations) approach one and the same limit L as P tends to zero

Formally, when do we right:
L = lim_{P→0} R(f,P,C) = lim_{P→0} ^{N}Σ_{i=1} f(c_{i})Δx_{i}
Th limit L is called the _____ _____ of f over [a,b]
If R(f, P, C)  L gets arbitrarily small as the norm P tends to zero, no matter how we choose the partition and sample points
definite integral

The notation ∫ f(x)dx was introduced by Leibniz in 1686. The symbol ∫ is an elongated S standing for ________. The differential dx corresponds to the length _____ along the xaxis

Define the Definite Integral
 The definite integral of f over [a,b], denoted by the integral sign, is the limit of Riemann sums
 ∫_{a}^{b} f(x)dx = lim_{P→0} ^{N}Σ_{i=1} f(c_{i})Δx_{i}
When this limit exists, we say that f is integrable over [a,b]

The definit integral is often called, more simply, the integral of f over [a,b]. The process of computing integrals is called _______ and f(x) is called the _______. The endpoints a and b of [a,b] are called the ______ ____ ______. Finally, we remark that any variable may be used as a variable of integration (this is a "dummy" variable). (pg 239)
 integration
 integrand
 limits of integration

Keep in mind that a Riemann sum R(f, P, C) is nothing more than an approximation to _____ based on _______, and that ∫_{a}^{b} f(x) dx is the number we obtain in the limit as we take ______ and ______ rectangles
 area
 rectangles
 thinner and thinner

Theorem 1: If f is continuous on [a,b], or if f is continuous with at most finitely many jump discontinuities, then f is ______ over [a,b]
integrable

When f(x) ≥ 0, then definite integral defines the area under the graph. To interpret the integral when f(x) takes on both positive and negative values, we define the notion of ______ _____, where regions below the xaxis are considered to have "______ area" Fig 4 (pg 230)

Signed area of a region formula:
Signed area of a region = (area above xaxis)  (area below xaxis)

Now observe that a Riemann sum is equal to the signed area of the corresponding rectangles:
R(f, C, P) = ?
R(f, C, P) = f(c_{1})Δx_{1} + f(c_{2})Δx_{2} +...+ f(c_{N})Δx_{N}

Indeed, if f(ci) < 0, then the corresponding rectangle lies below the xaxis and has signed area _________ Fig 5 (pg 240). What is the limit of the Riemann sums?
 f(ci)Δxi
 The signed area of the region between the graph and the xaxis

∫_{a}^{b }f(x)dx = ?
The limit of the Riemann sums = signed area of region between the graph and xaxis over [a,b]

In the rest of this section, we discuss some basic properties of definite integrals. First, we note that the integral of a constant function f(x) = C over [a,b] is the signed area _______ of rectangle as in Fig 7 (pg 240)
C(b  a)

State Integral of a Constant Theorem (2)
 For any constant C,
 ∫_{a}^{b} C dx = C(b  a)

State the Linearity of the Definite Integral Theorem
 If f and g are integrable over [a,b], then f + g and Cf are integrable (for any constant C), and
 ∫ab (f(x) + g(x))dx = ∫ab f(x)dx + ∫ab g(x)dx
 ∫ab Cf(x)dx = C ∫ab f(x)dx

∫_{0}^{b} x^{2}dx = ?
b^{3}/3

Reversing the the Limits of Integration (Definition)
 For a < b, we set
 ∫ab f(x)dx =  ∫ab f(x)dx

When a = b, the interval [a,b] = [a,a]has length zero and we define the definite integral to be _______

Definite integrals satisfy an important additivity property: If f is integrable and a ≤ b ≤ c as in Fig 9 (pg 242), then the integral from a to c is equal to the integral from:
a to b plus the integral from b to c

State the Additivity for Adjacent Intervals Theorem
 Let a ≤ b ≤ c, and assume that f is integrable. Then:
 ∫ac f(x)dx = ∫ab f(x)dx + ∫bc f(x)dx

State the Comparison Theorem
 If f and g are integrable and g(x) ≤ f(x) for x in [a,b], then
 ∫ab g(x)dx ≤ ∫ab f(x)dx

