5.2 The Definite Integral

  1. 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 = limN → ∞ RN = limN → ∞ LN = limN → ∞ MN
  2. 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
  3. Recall that RN, LN and MN 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)
  4. 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 <...<xN = b
    • Sample points: C = {c1,..., cN}: ci belongs to the subinterval [xi-1, xi] for all i = 1,...,N
  5. What is the length of ith subinterval [xi-1, xi]?
    Δxi = xi - xi-1
  6. Define the norm of P
    Norm: denoted ||P||, is the maximum of the lengths Δxi
  7. Given P and C, we construct the rectangle height ____ and base _____ over each subinterval [xi-1, xi], as in Fig 2B (pg 38) This rectangle has area _______ if f(c) ≥ 0. If f(ci) < 0, the rectangle extends below the x-axis, and f(ci)Δxi is the _______ of its area
    • f(ci)
    • Δxi
    • f(ci)Δxi 
    • negative
  8. Represent the Riemann sum in summation notation
    R(f,P,C) = NΣi=1 f(ci)Δxi = f(c1)Δx1 + f(c2)Δx2+...+ f(cN)ΔxN
  9. Keep in mind that RN, LN, and MN are particular examples of _______ _____ in which Δxi = ______ for all i, and the sample points ci are either _______ or _______
    • Riemann sums 
    • (b - a)/N
    • endpoints or midpoints
  10. 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
  11. Formally, when do we right: 
    L = lim||P||→0 R(f,P,C) = lim||P||→0  NΣi=1 f(ci)Δxi
    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
  12. 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 x-axis
    • summation
    • Δxi
  13. Define the Definite Integral
    • The definite integral of f over [a,b], denoted by the integral sign, is the limit of Riemann sums 
    • ab f(x)dx = lim||P||→0 NΣi=1 f(ci)Δxi 

    When this limit exists, we say that f is integrable over [a,b]
  14. 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
  15. Keep in mind that a Riemann sum R(f, P, C) is nothing more than an approximation to _____ based on _______, and that ∫ab f(x) dx is the number we obtain in the limit as we take ______ and ______ rectangles
    • area 
    • rectangles
    • thinner and thinner
  16. 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]
  17. 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 x-axis are considered to have "______ area" Fig 4 (pg 230)
    • signed area
    • negative area
  18. Signed area of a region formula:
    Signed area of a region = (area above x-axis) - (area below x-axis)
  19. 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(c1)Δx1 + f(c2)Δx2 +...+ f(cN)ΔxN
  20. Indeed, if f(ci) < 0, then the corresponding rectangle lies below the x-axis 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 x-axis
  21. ab f(x)dx = ?
    The limit of the Riemann sums = signed area of region between the graph and x-axis over [a,b]
  22. 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)
  23. State Integral of a Constant Theorem (2)
    • For any constant C,
    • ab C dx = C(b - a)
  24. 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
  25. 0b x2dx = ?
  26. Reversing the the Limits of Integration (Definition)
    • For a < b, we set 
    • ∫ab f(x)dx = - ∫ab f(x)dx
  27. When a = b, the interval [a,b] = [a,a]has length zero and we define the definite integral to be _______
    • zero 
    • aka ∫ab f(x)dx = 0
  28. 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
  29. 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
  30. 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
Card Set
5.2 The Definite Integral