The Definite Integral. Day 6 Motion Problems Strategies for Finding Total Area

Transcription

1 The Definite Integral Day 6 Motion Problems Strategies for Finding Total Area

2 ARRIVAL---HW Questions Working in PODS Additional Practice Packet p. 13 and 14 Make good use of your time! Practice makes perfect! Ask ME questions, ask your CLASSMATES questions!

3 Area Problems

4 Review Problem 1. Set up an integral that represents the shaded region. 2. Evaluate the integral using the Fundamental Theorem of Calculus THEN confirm your answer with fnint f ( x) = x ò dx = 9 x 2

5 Find the area of the shaded region f ( x) = x 2 Discuss with your partner what we need to do to find the area of the shaded region. Important to Remember: Integrals find area between the curve and the x-axis = 18

6 Remember.. When evaluating integrals, areas beneath the x-axis are negative. When evaluating total area, all areas are positive. a) Evaluate the integral 12 0 f ( x) dx f(x) f ( x) dx f ( x) dx f ( x) dx a) Find the total area of the graph f ( x) dx f ( x) dx f ( x) dx

7 Practice Find the area of the shaded region. NO CALCULATOR

8 YOU TRY!!

9 Check your answers

10 Today s Learning Outcomes Apply antiderivatives to motion problems Recognize the relationship between displacement and the total distance traveled by an object

11 Motion Problems

12 Remember. Given a position function: st Then Velocity: v t s ' t And Acceleration: a t v ' t s ''( t)

13 Given the velocity function, what could we determine. Given velocity v t Position s( t) + C= v t dt BECAUSE s(t) is the ANTIDERIVATIVE of v(t)

14 Remember our earlier example in this unit v( t) 55 mph from 2 : 00 to 5:00 Distance traveled = (55)(3) miles Graphically: What does this look like? OR Position of object at time t dt 55t 55(5 2) 55(3) miles 2 2

15 Terminology.. Displacement how far away from the starting point the object is at the end of a given time interval Distance Traveled amount of movement by the object in the positive and negative direction

16 The graph below shows the velocity of a particle over time. The area between the curve and the x-axis represents distance traveled. Area above the x-axis Distance traveled in the positive direction Movement away from the starting location Area below the x-axis Distance traveled in the negative direction Movement back towards the starting location

17 The graph shows the velocity of a particle over time. a) What is the displacement of the particle from 0 to 20 seconds? 1 1 a) = = 25 feet displacement 2 2

18 The graph shows the velocity of a particle over time. b) What is the total distance traveled from 0 to 20 seconds? 1 1 b) = = 53 feet total distance traveled

19 Another Example The velocity of a particle, in ft/sec, is given by v t 2 t. Find the displacement and total distance traveled from t 2 to 4. Before you start: Draw a sketch of the velocity curve to understand visually what is involved v t dt t 2 4 t Displacement is 12 feet. Velocity is positive in interval, therefore total distance also 12 feet.

20 The velocity of a particle, in ft/sec, is given by v t 2t 6. Find the displacement and total distance traveled from t 2 to 7. Before you start: Draw a sketch of the velocity curve to understand visually what is involved. 7 v t dt 7 2t t 6t Displacement is 15 feet. But what about total distance?

21 The velocity of a particle, in ft/sec, is given by v t 2t 6. Find the displacement and total distance traveled from t 2 to v t dt v t dt v t dt t 6t t 6t Total distance traveled is 17 feet.

22 How confident are you in your ability to Apply antiderivatives to motion problems Recognize the relationship between displacement and the total distance traveled by an object