Lesson 14.1

Just like a movie is a constantly moving figure, it can be broken into individual frames.

I may not be able to find the area of this figure (created with the x-axis), but I can estimate it by breaking it into rectangles.

Although this is a good estimate, I can have a better estimate using more rectangles.

As ∆x gets smaller, the my estimate gets more accurate. The integral is created by getting ∆x (the length of each rectangle) as small as possible.

Velocity-time curve: a graph in which x is the velocity and y is the time.

• A car accelerates from 0 to 60 ft/sec (41 mi/hour) in 8 seconds with a velocity

after t seconds. Estimate how far the car travels in this time

by dividing the interval 0 ≤ t ≤ 8 into 4 subintervals and using the midpoints of those intervals.

sec60)8(

16

15)( 2 ft

ttfv

• Now find the area of each of the rectangles!(remember – use midpoint to find the height)

Area: length * heightBlue : 2 * f(1)General: 2 * f(1) + 2 *f(3) + 2 * f(5) + 2 f(7)

2 * 14.0625+ 2 *36.5625 + 2 * 51.5625+ 2 * 59.0625

322.5

sec60)8(

16

15)( 2 ft

ttfv

Calculate how far the car in Example 1 travels if the velocity is changed to 7.5t after t seconds.

A = ½ bhA = ½ * 8 *

60A = 240

feet

(8, 60)

2, 4, 5, 7, 8