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.

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Presentation transcript:

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

Example 1 Evaluate the definite integral

Example 2 Evaluate the definite integral

Example 3 Evaluate the definite integral

Example 4 Evaluate the definite integral

Example 5 Evaluate the definite integral

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

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

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.