Ccc83 8.1 May - practice PDF

Preview

Summary

practice...

Description

CHAPTER 8 Using the Definite Integral In this chapter we explore more applications of the definite integral by using it to measure distance traveled, volume of solids, and the average value of a function. The problem solving strategy employed in these applications is similar to that used to find areas under curves. Given a continuous function f on an interval [a, b], we start by dividing the interval into n equal subdivisions, each of length ∆x = ( b − a) / n . Then we choose a single input x from the subinterval and use the corresponding output f ( x ) as the approximate constant value of f over each subinterval. If the quantity we are measuring is approximated by the product f ( x )⋅ ∆ x for each subinterval, then we add up all of the products to form an approximating Riemann sum. The limiting value of this Riemann sum is a definite integral that measures the desired quantity. 8.1

Net and Total Distance Traveled

In Chapter 3 we discovered that distance traveled is the definite integral of velocity. Consider an object moving along a straight line with velocity v(t ) at time t . If v(t ) ≥ 0 , then b

∫a v(t ) dt gives the distance traveled during the time interval a ≤ t ≤ b . In the figure below the

area of each rectangle is v( ti ) ∆t . If the velocity were constant this area would represent the distance traveled by the object during the time period ∆t . Because the velocity is not constant, the shaded area only approximates the distance traveled for this time period. v

velocity

Area = v(t i ).∆ t

a

ti ∆t

b

time

Summing the areas of all the rectangles we get v( t1)⋅ ∆ t + v( t 2)⋅ ∆ t + ... + v(tn )⋅ ∆ t , an approximation to the total distance traveled between t = a and t = b . As the number, n, of subintervals increases, the length of each subinterval, ∆t , approaches 0 and the sums approach the area under the graph of the velocity function, which represents the total distance traveled. Example 1: Finding Distance Traveled When Velocity Varies During a trip a car's speedometer reading, in miles per hour, is given by the function v where v(t ) = 25 − 25 cos(2 πt ) from t = 0 to t = 1 hours. Sketch a graph of the velocity function and determine how far the car traveled during the trip.

373

374

Chapter 8 - Using the Definite Integral

Solution:

8.1

v v( t ) = 25 − 25 cos(2 πt )

The total distance traveled during the trip is given by the area under the curve from t = 0 to t = 1. Since v(t ) ≥ 0 for 0 ≤ t ≤ 1, we have 1

∫0

1

∫0

v (t ) dt = [ 25 − 25 cos( 2πt)] dt ⎤ 25 = ⎡25t − 2 sin(2πt ) ⎢⎣

π

1

⎥⎦ 0

= ( 25) − ( 0) = 25. The car travels 25 miles from t = 0 to t = 1. Integrating velocity over a time interval [a, b] gives the change in position of the moving object over the time interval. We should think of this as the net distance traveled. For example, suppose you walk 5 steps forward and then take 3 steps backward; the net change in your position is 2 steps forward, even though you have traveled a total distance of 8 steps. Start

5 steps 3 steps

Net Distance = 2 steps

The direction of movement is indicated by the sign of the velocity. A negative velocity indicates movement in the opposite direction of a positive velocity. Thus, if v(t ) is sometimes negative, then b

∫a v(t ) dt measures the net distance from the starting position to the ending position. To get the b total distance traveled by an object during a time interval [a, b] , we must calculate ∫ v(t ) dt . a net distance traveled

=

b

∫a v(t) dt b

total distance traveled =

∫ a v(t )

dt

Example 2: Finding Net and Total Distance Traveled An object moves along a coordinate line with distance to the right considered positive. If the 2

velocity of the object is given by v(t ) = t − 7t + 10 over the time interval 1 ≤ t ≤ 3 , find the net and the total distance traveled.

8.1

Chapter 8 - Using the Definite Integral

375

Solution: The graph of the velocity function is shown in the figure

Graph of the velocity function

The direction the object is moving at any time t is determined by the sign of the velocity v(t ). In this example, the velocity is positive for 1 ≤ t < 2 and the object is moving to the right during this time. The velocity is negative for 2 < t ≤ 3 and the object is moving to the left during this time. The object reverses direction at t = 2 when the velocity is zero. The net distance traveled by the object is measured by the net area of the region bounded by the graph and the t–axis over the time interval [1, 3]. 3 ⎡t 3 7 2 ⎤ 32 + 30) − (1 − 72 +10 ) = 2 ( t − 7 t + 10) dt = ⎢ 3 − t + 10t ⎥ = ( 9 − 63 2 2 3 3 1 ⎥⎦ ⎣⎢ 1

∫

This means the object moves a net distance of 2/3 units to the right. The total distance traveled is 2 3 3 2 t − 7t + 10 dt = ( t 2 − 7t + 10) dt + ( −t 2 + 7t − 10 ) dt 1 2 1 2 3 ⎤ ⎡t 3 7 2 ⎤ ⎡ t3 7 2 = ⎢ 3 − t + 10t ⎥ + ⎢− 3 + t − 10t ⎥ 2 2 ⎥⎦ ⎢⎣ ⎥⎦ 2 ⎢⎣ 1 = 11 + 7 = 3 . 6 6

∫

∫

∫

We see from this calculation that the object moved 11 units to the right, reversed direction, 6 and then moved 67 units to the left for a total distance of 3 units and a net distance of 2 units 3 to the right.

376

8.1

Chapter 8 - Using the Definite Integral

8.1

Exercises

In Exercises 1–6 the velocity function (in feet per second) is given for an object moving along a line. Find a) the net distance traveled and b) the total distance traveled by the object during the given time interval. 1.

v(t ) = 4 − 2 t,

0 ≤t ≤3

2.

v (t ) = t2 − 6 t − 8,

1 ≤ t ≤5

3.

v (t ) = 2 t 2 − 6 t + 4,

0 ≤t ≤ 4

4.

v(t ) = t cos t ,

0 ≤t ≤π

5.

v (t ) = sec2 t − 2,

0≤t≤

6.

v( t) = 5 − e0. 5t ,

1 ≤t ≤4

π 3

In Exercises 7–10, first find an equation for the velocity of a moving object from the equation of the acceleration. Then find the net and total distance traveled by the moving object in the given time interval. 7.

a(t ) = 6 t − 18 ft/sec2, v(0 ) = 24 ft/sec, from t = 1 to t = 5 sec

8.

a( t) = t ft/sec2, v(0 ) = −6 ft/sec, from t = 0 to t = 9 sec

9.

a(t ) = −2π cos( 2π t) ft/sec2, v(0 ) = 1 ft/sec, from t = 0 to t = 1 sec

10.

a( t) = 3 cm/sec2, v(0 ) = 2 cm/sec, from t = 0 to t = 1 sec

11.

The velocity of a car was read from its speedometer at ten second intervals and recorded in the table. Use the Trapezoid Rule with six equal subdivisions to estimate the distance traveled by the car. t (sec) 0 10 20 30 40 50 60 v (ft/sec) 0 38 42 48 51 50 45

12.

A particle moves along a line so that at any time t > 0 its acceleration is given by a(t ) = ln t ft/sec2. At time t = 1 sec the velocity of the particle is v(1) = –2 ft/sec. a) Write an expression for the velocity of the particle.

t+1

b) For what values of t is the particle moving to the right? c) What is the minimum velocity of the particle? d) Find the total distance traveled by the particle from t = 2 to t = 4 seconds.

8.1

13.

377

Chapter 8 - Using the Definite Integral

A car is moving forward and backward along a straight road from A to B, π starting from A at time t = 0. The car's velocity is given by v( t) = 1 + 2sin( t ) 6

where t is in minutes and v is in km/min. The graph of the velocity function is given below. v (km/min) 3 2 1

t (min) 2

4

6

8

10

12

-1 -2 -3

a) What is the velocity of the car at t = 0 ? b) At what time(s) t does the car change direction? c) Use a rectangle summing program (e.g., RSUM) to estimate the car's distance from A at t = 9 . d) Find the average velocity of the car between t = 0 and t = 9 . 14.

A particle moves along a line so that at any time t ≥ 0 its velocity is given by v(t ) = At time t = 0 the position of the particle is s(0 ) = 5. a) Determine the maximum velocity of the particle. Justify your answer. b) Determine the position of the particle at t = 3. c) What is the total distance traveled by the particle from t =1 to t = 3? d) Find the limiting value of the velocity as t increases without bound.

15. A car is moving along a straight road from A to B, starting from A at time t = 0 . Below is a graph of the car's velocity (positive direction from A to B), plotted against time. 2 1 v(km/min)

1

2

3

4

5

6

7

8

–1 –2 time (minutes)

a) How many kilometers away from A is the car at time t = 6 ? b) At what time does the car change direction? Explain briefly. c) Sketch a graph of the acceleration of the car.

t 1 +t

2

....