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Areas and Definite Integrals

Areas and Definite Integrals. Objectives. Students will be able to Calculate a definite integral. Calculate the area between a curve and the x -axis over a given interval. Use definite integrals to application problems. Fundamental Theorem of Calculus.

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Areas and Definite Integrals

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  1. Areas and Definite Integrals

  2. Objectives Students will be able to • Calculate a definite integral. • Calculate the area between a curve and the x-axis over a given interval. • Use definite integrals to application problems.

  3. Fundamental Theorem of Calculus For a continuous function f(x) over an interval [a, b], let F be any antiderivative of f. Then

  4. Example 1 Evaluate the definite integral

  5. Example 2 Evaluate the definite integral

  6. Example 3 Evaluate the definite integral

  7. Example 4 Evaluate the definite integral

  8. Example 5 Evaluate the definite integral

  9. Example 6 Find the area between and the x-axis from x = 2 to x = 4.

  10. Example 7 The graph of f(x) below consists of two straight line segments and two quarter circles. Find the value of

  11. Example 8 An oil tanker is leaking oil at the rate of (in barrels per hour), where t is the time (in hours) after the tanker hits a hidden rock (when t = 0). a. Find the total number of barrels that the ship will leak on the first day. b. Find the total number of barrels that the ship will leak on the second day.

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