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Chapter 7 Additional Integration Topics

Chapter 7 Additional Integration Topics. Section 1 Area Between Curves. Learning Objectives for Section 7.1 Area Between Two Curves. The student will be able to Determine the area between a curve and the x axis , Determine the area between two curves, and

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Chapter 7 Additional Integration Topics

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  1. Chapter 7Additional Integration Topics Section 1 Area Between Curves

  2. Learning Objectives for Section 7.1 Area Between Two Curves • The student will be able to • Determine the area between a curve and the x axis , • Determine the area between two curves, and • 3. Solve an application involving income distribution.

  3. Negative Values for the Definite Integral If f (x) is positive for some values of x on [a, b] and negative for others, then the definite integral y = f (x) is the cumulative sum of the signed areas between the graph of f (x) and the x axis. Areas above the x axis are positive, and areas below are negative. In this section we want find the (unsigned) actual area between a curve and the x axis or between two curves. This is never negative! B a A b

  4. Negative Values The (unsigned) area between the graph of a negative function and the x axis is equal to the definite integral of the negative of the function. y = f (x) B a A b c

  5. Area Between Curve and x axis The area between f (x) and the x axis can be found as follows: For f (x) > 0 over [a, b] (i.e. f (x) is above the x axis): Area = For f (x) < 0 over [a, b] (i.e. f (x) is below the x axis): Area =

  6. A 1 A 2 Example Find the area bounded by y = 4 – x2; y = 0, 1 ≤ x ≤ 4.

  7. A 1 A 2 Example Find the area bounded by y = 4 – x2; y = 0, 1 ≤ x ≤ 4. Area = A1 – A2

  8. Area Between Two Curves Theorem 1. If f (x) ≥ g(x) over the interval [a, b], then the area bounded by y = f (x) and y = g(x) for a ≤ x ≤ b is given by f (x) A g (x) a b

  9. Example Find the area bounded by y = x2 – 1 andy = 3.

  10. Example Find the area bounded by y = x2 – 1 andy = 3. Note the two curves intersect at –2 and 2, and y = 3 is the larger function on –2 ≤ x ≤ 2.

  11. Computing Areas Using a Numerical Integration Routine Suppose we want to find the area bounded by

  12. Computing Areas Using a Numerical Integration Routine Suppose we want to find the area bounded by First, we use a graphing calculator to graph the functions f and g and find their intersection points as in the figure. We see that the graph of f is bell shaped and the graph of g is a parabola, and that f (x) >g(x) on the interval [–1.131, 1.131].

  13. Computing Areas Using a Numerical Integration Routine Then we compute the area by a numerical integration routine on a calculator.

  14. Application: Income Distribution The U.S. Bureau of the Census compiles data on distribution of income among families. This data can be fitted to a curve using regression analysis. This curve is called a Lorenz curve. The variable x represents the cumulative percentage of families at or below the given income level. The variable y represents the cumulative percentage of total family income received. If we have absolute equality of income (every family has the same income), the Lorenz curve is y = x.

  15. Application(continued) Curve of absolute equality y = x For example, data point (approximately) (0.4, 0.09) in the table indicates that the bottom 40% of families receive only 9% of the total income for all families. Data point (.6, .26) indicates that the bottom 60% of families receive only 26% of the total income, etc. Lorenz Curve (0.4, 0.09)

  16. Application(continued) If we have absolute inequality of income (one family has all the income, the rest have none), the Lorenz curve is y(x) = 0 for x < 1, y(1) = 1. The maximum possible area between the Lorenz curve and the line y = x is ½, which occurs in the case of absolute inequality. That is, the area between the Lorenz curve and the line y = x is between 0 and ½. The Gini Index is two times the area between the Lorenz curve and the line y = x. It can take on values between 0 and 1.

  17. Application(continued) Gini Index of Income Concentration If y = f (x) is the Lorenz curve, then Index of income concentration = A measure of 0 indicates absolute equality. A measure of 1 indicates absolute inequality.

  18. Example A country is planning changes in tax structure in order to provide a more equitable distribution of income. The two Lorenz curves are: f (x) = x2.3 currently, and g(x) = 0.4x + 0.6x2 proposed. Will the proposed changes work?

  19. Example(continue) Currently: Gini Index of income concentration = Future: Gini Index of income concentration = The Gini index is decreasing, so the future distribution will be more equitable.

  20. Summary • We reviewed the definite integral as the area between a curve and the x axis. • We learned how to calculate area when the curve was below the x axis. • We learned how to calculate the area between two curves. • We learned about the Lorenz curve and the Gini index of income concentration. Barnett/Ziegler/Byleen Business Calculus 12e

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