Chapter 3 Review

Chapter 3: Accumulated change: the definite integral



• SECTION 3.1: Accumulated change
  
  
  1. Suggested Problems: #5, 6, 7, 8, 9.
     
  2. The accumulated change in a function f(x) over an interval [a, b] is given by f(b) - f(a).
     
  3. Be able to approximate the accumulated change in a function f(x) over an interval [a, b] given only a table of values of the derivative function f(x).
     
  4. Be able to approximate the accumulated change in a function f(x) over an interval [a, b] given only a graph of the derivative function f(x).
     
  5. Be able to approximate the accumulated change in a function f(x) over an interval [a, b] given only a formula for the derivative function f(x).
     
  6. If f(x) = c is constant over the interval [a, b], then f(b) - f(a) = c (b - a).
     
  7. Given the rate of change f(x), the units of the accumulated change in f(x) over an interval [a, b] are given by (units of f(x))*(units of x).
     
  8. Be able to graphically represent accumulated change in a function f(x) using areas of rectangles under the graph of f(x).
     
  9. Approximations that utilize smaller intervals improve the accuracy of the approximation for accumulated change.
     
  10. Be able to give upper estimates and lower estimates of the accumulated change in a function given a derivative that is either increasing or decreasing.
      
  
  
  
• SECTION 3.2: The definite integral
  
  
  1. Suggested Problems: #3, 4, 10, 13, 17, 20, 21.
     
  2. Be able to produce underestimates and overestimates of the total change in a quantity, given a rate of change, using sums of areas of rectangles, provided the rate of change in a quantity is either increasing or decreasing.
     
  3. Be able to produce right sum approximations and left sum approximations of the total change in a quantity, given a rate of change, using sums of areas of rectangles.
     
  4. If f(x) is an increasing function, then a right sum approximation of f(x) dx yields an upper estimate of f(x) dx, and a left sum approximation of f(x) dx yields a lower estimate of f(x) dx.
     
  5. If f(x) is a decreasing function, then a right sum approximation of f(x) dx yields a lower estimate of f(x) dx, and a left sum approximation of f(x) dx yields an upper estimate of f(x) dx.
     
  6. We can improve the approximation of the total change in a quantity over an interval [a, b] by increasing the number n of subintervals into which we divide [a, b], which is equivalent to decreasing the size of x = (b - a)/n.
     
  7. Be familiar with the sigma notation ( notation) for the representation of Riemann sums.
     
  8. Be able to approximate the definite integral of a function given a table of values of the function.
     
  9. Be able to approximate the definite integral of a function given a graph of the function.
     
  10. Be able to approximate the definite integral of a function given a formula for the function.
      
  11. Be able to use a calculator to compute the definite integral of a function given a formula of the function.
      
  12. The total change in a quantity f(x) over an interval [a, b] is given by the area underneath the graph of f(x) between x = a and x = b (assuming that f(x) > 0).
      
  13. If f(x) is continuous on the interval [a, b], then the definite integral of f(x) from x = a to x = b, written f(x) dx, is the limit of a Riemann sum approximation having n subdivisions of [a, b] as n becomes arbitrarily large.
      
  14. With respect to the definite integral f(x) dx, the function f(x) is referred to as the integrand, and the values a and b are referred to as the limits of integration.
      
  
  
  
• SECTION 3.3: The definite integral as area
  
  
  1. Suggested Problems: #2, 3, 7, 8, 9, 15, 17.
     
  2. When a function f(x) is positive over an interval [a, b], the definite integral f(x) dx yields the area of the region below the graph of y = f(x) and above the x-axis from the value x = a to the value x = b.
     
  3. When the function f(x) is positive for some values of x and negative for other values of x over the interval [a, b], the definite integral f(x) dx yields the sum of the areas of the regions between the graph of y = f(x) and the x-axis, when f(x) is above the x-axis, minus the sum of the areas of the regions between the graph of y = f(x) and the x-axis, when f(x) is below the x-axis, from the value x = a to the value x = b.
     
  4. If f(x) and g(x) are two functions with f(x) g(x) for all x in the interval [a, b], then the area between the graphs of f(x) and g(x) (which would be above the curve y = g(x) and below the curve y = f(x)) is given by the definite integral (f(x) - g(x)) dx.
     
  5. Be able to represent the definite integral graphically.
     
  6. Be able to find the area between two curves.
     
  
  
  
• SECTION 3.4: Interpretations of the definite integral
  
  
  1. Suggested Problems: #1, 5, 7 , 9, 13.
     
  2. The units of f(x) dx are given in (the units of f(x))*(the units of x).
     
  3. If f(t) represents a rate of change of some quantity, then the total change in that quantity from t = a to t = b is given by f(t) dt.
     
  4. If y = f(x) is positive over [a, b], then the area below the graph of y = f(x) and above the x-axis is given by f(x) dx.
     
  5. If the rate of change of two quantities is given graphically, be able to determine which quantity has the greatest total change over a given interval.
     
  
  
  
• SECTION 3.5: The fundamental theorem of calculus
  
  
  1. Suggested Problems: #1, 2, 7, 10, 11, 15.
     
  2. The Fundamental Theorem of Calculus: Let F(t) be a function such that F(t) exits and is continuous for a t b. Then
     F(t) dt = F(b) - F(a).
  3. The definite integral f(t) dt is related to the anti-derivative of the function f(t). If we know that f(t) is the derivative of another function F(t), then f(t) dt = F(b) - F(a).
     
  4. Given the marginal cost function C(q), we can find the total change in the cost function over some interval [a, b], which is C(b) - C(a), by
     C(q) dq = C(b) - C(a), regardless of whether or not we know the cost function itself.
     
  
  
  

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Last modified: Fri Apr 12 2002