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The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus

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Introduction to The Fundamental Theorem Of Calculus - At A Glance:

When Velocity is Non-Negative

Again, let's assume we're cruising on the highway looking for some gas station nourishment. We could actually figure out the distance we've travelled by knowing our velocity and the time we spent searching.

We can only use the formula

distance = velocity × time

if velocity is constant on the time interval we're looking at. If we graph time on the horizontal axis and (constant) velocity on the vertical axis, we get this picture:

The area of this rectangle is velocity × time, which equals distance. The area of this rectangle also happens to be the definite integral of the (constant) velocity function on [a,b]. In symbols, when velocity is constant and positive on [a,b], the distance travelled from t = a to t = b is

Sample Problem

A car is traveling at v(t) = 60 mph. How far does the car travel in 20 minutes?

Answer.

20 minutes is one third of an hour, so

This is the area between the graph of the constant function v(t) = 60 and the t-axis on the interval (or on any other interval of length  hour).

If velocity isn't constant it's not clear how to apply the formula

distance = velocity × time.

What do we use for velocity? One approach is to pretend the velocity is constant: pick one reasonable value for the velocity and pretend that's the velocity for the whole time interval.

If we want to get a better estimate of distance travelled, we can split up the time interval into sub-intervals and pretend that velocity is constant on each sub-interval.

Sample Problem

Suppose Jen's velocity in mph was measured every ten minutes for one hour, and that her velocity was decreasing over that hour. The recorded values are shown in the table below. Estimate how far Jen travelled during the hour.

Answer.

We don't know how fast Jen was going at every minute of the hour, but we can pretend that she was going 60 mph for the first ten minutes. Similarly, we can pretend she was going

55 mph from t = 10 to t = 20 minutes

50 mph from t = 20 to t = 30 minutes

40 mph from t = 30 to t = 40 minutes

35 mph from t = 40 to t = 50 minutes

25 mph from t = 50 to t = 60 minutes.

In real life her velocity would probably look something like this

but to make the problem easier we're pretending her velocity looks like this:

Using the formula

distance = velocity × time

on each ten-minute sub-interval, we estimate that

Adding up the estimated distance for each 10-minute sub-interval, we estimate that over the full hour Jen travelled

10 + 9.2 + 8.3 + 6.7 + 5.8 + 4.2 = 44.2 miles.

On each sub-interval we're overestimating Jen's velocity - she wasn't going 60 mph for all of the first ten minutes - so this answer is an overestimate of how far Jen actually travelled.

In the example above, what we really did was use a left-hand sum with 6 sub-intervals to estimate

The upper limit of integration is 1 because the function v(t) expects t to be in hours.

When we use values of the velocity function and a right- or left-hand sum to approximate distance travelled from time t = a to time t = b, we're also approximating the integral of the velocity function on [a,b].

If the velocity function looks like this:

and we pretend it looks like this:

then the area of the rectangle on each sub-interval is our estimate of how far the cheetah or snail or whatever has travelled during that sub-interval.

As our velocity measurements get closer together, our estimates approach the real distance travelled. Our estimates are also approaching the integral of the velocity function. Since our estimates can't approach two different things, the real distance travelled and the integral of the velocity function must be the same. In symbols, when v(t) is non-negative,

Example 1

Suppose JoJo was driving 30 mph at 5 pm and 40 mph at 7 pm. How far did she travel between 5 and 7 pm?


Example 2

The velocity of a racing snail, in feet per second, is given below.

How far does the racing snail travel

(a) from t = 0 to t = 10 seconds?

(b) from t = 10 to t = 15 seconds?

(c) from t = 15 to t = 25 seconds?

(d) from t = 0 to t = 25 seconds?


Example 3

A bug moves along a numberline with velocity v(t) in units per second given by the graph below.

If the bug is at 3 on the numberline when t = 0 seconds, where is the bug when

(a) t = 3 seconds?

(b) t = 5 seconds?


Exercise 1

Suppose Jen's velocity in mph was measured every ten minutes for one hour, and that her velocity was decreasing over that hour. The recorded values are shown in the table below. Use a right-hand sum to estimate how far Jen travelled during that hour. Is this an over- or under-estimate of the distance she really travelled?

Exercise 2

The table below gives the velocity of a tortoise over nine minutes with measurements taken every 3 minutes.

Estimate the distance the tortoise travels during these nine minutes, using

a) a left-hand sum

b) a right-hand sum

c) a trapezoid sum

Exercise 3

The table below gives the velocity of a dinosaur over thirty seconds, with measurements taken every 5 seconds.

Estimate the distance the dinosaur travels during these thirty seconds, using

a) a left-hand sum

b) a right-hand sum

c) a trapezoid sum

Exercise 4

The velocity of a garden snail over 3 minutes, measured twice per minute, is given in the table below.

Estimate the distance the snail travels during these three minutes.

Exercise 5

The velocity of a cheetah over one minute, measured every 10 seconds, is given in the table below.

Estimate the distance the cheetah travels during this minute.

Exercise 6

Use the trapezoid rule and the table below to estimate how far the cheetah travels in one minute.

Exercise 7

The velocity of a pink elephant, in feet per minute, is given by the function v(t) below.

How far does the elephant travel

(a) from t = 0 to t = 10?

(b) from t = 10 to t = 25?

(c) from t = 25 to t = 30?

(d) from t = 0 to t = 30?

Exercise 8

A bug starts out at position -4 on a number line and crawls along with velocity v(t) units per second, where v(t) is given by the graph below.

Where is the bug when

(a) t = 2?

(b) t = 3?

(c) t = 5?

Exercise 9

A cat is climbing a tree. The cat is one foot above ground when its owner starts a stopwatch. The cat's velocity, in feet per second, is given by the following graph:

How high in the tree is the cat when the stopwatch reads

(a) 3 seconds?

(b) 6 seconds?

(c) 8 seconds?

(d) 10 seconds?

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