• We consider how to compute areas under curves.

During the morning commute, a car drives on a highway at a steady speed of \(90\) km/h from 7:30 AM to 9:00 AM. What is the total distance travelled by the car during that time?

Solution

It is well known that

as long as the speed is constant.

Thus, the answer is \(90\) km/h \(\times\) \(1.5\) h \( = \) \(135\) km.

During the morning commute, a car drives on a highway at a steady speed of \(90\) km/h from 7:30 AM to 9:00 AM. What is the total distance travelled by the car during that time?

Solution

Notice that distance can be interpreted as the area under the curve, \(v(t)\); it is a rectangle with base \(1.5\) and height \(90\).

How do we find the total distance travelled if the car's speed varies over the duration of the trip?

Is it still the area under the curve, \(v(t)\), over the given time interval?

Estimate the total distance travelled by the car in \(35\) minutes.

Attempt 1

Notice the estimated total distance is exactly the area of the rectangle whose base is the time interval \(35\) minutes (or \(\frac{35}{60}\) hours) and whose height is \(48\) km/h, the estimated average velocity of the car.

Estimate the total distance travelled by the car in \(35\) minutes.

Attempt 1

Remarks

Recall,

Since we do not know the total distance travelled, we cannot calculate the average velocity exactly.

Since our estimatation of the average velocity was formed using only \(7\) discrete points, we have no way of knowing how close our estimation is to the true average velocity.

Thus, we have no way of knowing the accuracy of our estimation of the total distance travelled.

Estimate the total distance travelled by the car in \(35\) minutes.

Attempt 2

For our second attempt, let's divide the trip into \(5\) minute time intervals, and assume that the car was travelling at a constant velocity during each \(5\) minute interval.

Assume that the car was travelling at a constant velocity of \(v(5)=50\) for the entire time interval \([0,5]\).

Therefore, the estimated total distance travelled from \(t=0\) to \(t=5\) is \(50\) km/h \(\times \frac{5}{60}\) h \(\approx 4.17\) km.

Assume a constant velocity of \(v(10)=53\) on the entire interval \((5,10]\).

Therefore, the car travels \(53\) km/h \(\times \frac{5}{60}\) h \(\approx 4.42\) km.

Estimate the total distance travelled by the car in \(35\) minutes.

Attempt 2

Estimate the total distance travelled by the car in \(35\) minutes.

Attempt 2

Graphically, we think of velocity as constant over a \(5\) minute interval.

The total distance can be interpreted as the sum of the areas of seven rectangles.

Each rectangle has base \(\frac{5}{60}\), but they have varying heights, each equal to the velocity recorded during that time period.

Estimate the total distance travelled by the car in \(35\) minutes.

Wrap Up

Both attempts involve estimations of average velocities over time intervals using our discrete data points.

Attempt 1

Attempt 2

We don't know which attempt provides the best estimate for the distance of the trip.

If we take more and more readings, then our estimatations, using the same procedures, should become more and more accurate.

Estimate the total distance travelled by the car in \(35\) minutes.

Wrap Up

What if we told you that our \(7\) data points actually lie on a continuous curve representing the function \(v(t)\)?