Chapter 6.2, Problem 46E

To determine

Quantity not represented by the definite integral.

Answer to Problem 46E

The quantity of distance traveled by a train moving at 70 mph for 8 minutes cannot be represented by the definite integral.

Explanation of Solution

Given information:

  ${\displaystyle {\int }_0^{8}70dt}$

While integrating ${\displaystyle {\int }_a^{b}f\left(x\right)dt}$ ,

The units of a and b must be same as the units for x .

This implies

While integrating ${\displaystyle {\int }_0^{8}70dt}$ ,

The units of 0 and 8 must be same as the units for t .

For the quantity of distance traveled by a train moving at 70 mph for 8 minutes:

  $f\left(t\right)=70$ has the units of hours for t .

Since 70 represents miles per hour, but 0 and 8 have units of minutes.

For the quantity of volume of ice cream produced by a machine making 70 gallons per hour for 8 hours:

  $f\left(t\right)=70$ has units of gallons per hour.

Thus,

t also has units of hours.

The Problem requires the volume produced in 8 hours.

Then

The bounds of 0 and 8 also have units of hours.

For the quantity of length of track left by a snail traveling at 70 cm per hour for 8 hours:

  $f\left(t\right)=70$ has units of cm per hour.

Thus,

t also has units of hours.

The Problem requires the length of a track for 8 hours.

Then

The bounds of 0 and 8 also have units of hours.

For the quantity of total sales of a company selling $70 of merchandise per hour for 8 hours:

  $f\left(t\right)=70$ has units of merchandise per hour.

Thus,

t also has units of hours.

The Problem requires the total sales for 8 hours.

Then

The bounds of 0 and 8 also have units of hours.

For the quantity of the amount the tide has risen 8 minutes after low tide if it rises at a rate of 70 mm per minute:

  $f\left(t\right)=70$ has units of mm per minute.

Thus,

t also has units of minutes.

The Problem requires the amount of tide has risen in 8 minutes.

Then

The bounds of 0 and 8 also have units of minutes.

Conclusion:

We can conclude that in option (A), 70 represents miles per hour, but 0 and 8 have units of minutes.

Thus,

The quantity of distance traveled by a train moving at 70mph for 8 minutes cannot be represented by ${\displaystyle {\int }_0^{8}70dt}$ .

Therefore,

The correct answer is option (A).