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Limits and Motion: The Area Problem Today’s Objective: I can calculate definite integrals using area.

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Presentation on theme: "Limits and Motion: The Area Problem Today’s Objective: I can calculate definite integrals using area."— Presentation transcript:

1 Limits and Motion: The Area Problem Today’s Objective: I can calculate definite integrals using area.

2

3 Limits at Infinity

4 An automobile travels at a constant rate of 48 miles per hour for 2 hours and 30 minutes. How far does the automobile travel? Distance = Area under the curve What if the speed varies?

5 Right rectangular approximation method Left rectangular approximation method

6 BaseHeightArea 1/21/40.125 1/210.500 1/29/41.125 1/242.000 1/225/43.125 1/294.50 Total Area 11.375

7 Definite Integral Find Find the area under the curve. Shape is a trapezoid. = area under the curve

8 Suppose a ball rolls down a ramp so that its velocity after t seconds is always 2t feet per second. How far does it fall during the first 3 seconds? Distance traveled = area under the velocity graph Integral notation:

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