The Fundamental Theorem of Calculus Exercises

Answer

First we find the value of the integral:

To explain what the integral represents in the language of the problem, we need to know units for t and units for the integral. We're told that t is measured in hours and v(t) in miles per hour. This means the integral is measured in miles. So

is the distance Bob drove from time t = 0 hours to time t = 2 hours. Since we're told 3pm corresponds to t = 0, t = 2 corresponds to 5pm. Putting everything together,

is the distance Bob drove from 3pm to 5pm.

Answer

We want to know the change in the amount of water that has come out of the faucet from time t = 0 minutes to time t = 2 minutes. This means we need to integrate v(t) from 0 to 2.

Rounding to the nearest gallon, 11 gallons come out of the faucet during the first 2 minutes it is turned on.

Answer

This question is asking for the difference between p(100) and p(1000). We don't even need an integral for this.

As one moves from 100m to 1000m above sea level, the air pressure drops by approximately 10,255 Pascals.

We could use an integral if we wanted. Since we're being asked for the change in pressure between h = 100 and h = 1000, we need to integrate the rate of change of pressure, p'(h), from 100 to 1000. However, using the FTC this gets us the same expression we had when we didn't bother to use the integral:

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Since h is given in feet and v(h) in decibels per feet, the integral

is measured in decibels. This integral represents the change in Fluffles' volume as she climbs from 10 feet up in the tree to 20 feet up in the tree. It remains to evaluate the integral.

Fluffles' volume changes by approximately 8 decibels as she climbs from 10 feet up in the tree to 20 feet up in the tree.

Answer

We need to figure out how many square feet Sonya can paint in 3 hours. Then for each office we need to determine how many square feet there are to paint, and see if this is greater or smaller than the number of square feet Sonya can paint in 3 hours.

To find out how many square feet Sonya can paint in 3 hours, we integrate her rate of painting from time t = 0 hours to time t = 3 hours:

The first office is 12 by 12 feet with a height of 8 feet. Since each wall has width 12 and height 8, and there are 4 walls, the office has

12 × 8 × 4 = 384

square feet of wall that need to be painted. Since 384 is less than 432, Sonya can paint the walls of this office in 3 hours.

The second office is 14 x 14 feet with a height of 8 feet. Since each wall has width 14 and height 8, and there are 4 walls, the office has

14 × 8 × 4 = 448

square feet of wall that need to be painted. Since 448 is greater than 432, Sonya cannot paint the walls of this office in 3 hours.

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The function c(p) is the rate of change of cost with respect to weight. This means

is the change in cost from p = 20 to p = 50, or the change in cost as the bag goes from 20 to 50 pounds. For p ≤ 50 we have c(p) = 0.5, so

As a bag's weight increases from 20 pounds to 50 pounds, the cost to check it increases by

Answer

The distance the ball travels during the first 2 seconds after it is thrown is given by

Since we know this integral must be equal to 25, we need to solve the equation

for v₀.

We conclude that the ball's initial speed was 2.7 meters per second.

Answer

We want to find a depth H in feet such that the change in p from h = 15 to h = H is 800 pcf.

In symbols, we want to find H for which

Here goes:

Rounding to the nearest foot, if one dives to a depth of 28 feet (13 feet deeper since we're already at a depth of 15 deep), the water pressure will increase by 800 pcf.

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To find the number of square feet Shauna paints between 2 and 3pm, we integrate her rate of painting from t = 2 to t = 3:

We see Shauna painted  square feet between 2pm and 3pm. Since the problem says she painted 100 square feet during this time interval, we must have

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We want to find how many hours H it takes Tom to paint 800 square feet. In H hours Tom can paint

square feet, so we want to find H such that

Here we go:

Rearranging, we get

2H² – 200H + 800 = 0

and dividing through by 2,

H² – 100H + 400 = 0.

It's not obvious whether we can factor this, but we can use the quadratic formula to solve for H:

The two answers are

and

Since 95 hours isn't a realistic amount of time to spend painting a fence, we'll throw that answer out and keep the answer that's approximately 4 hours. The amount 0.174 hours is approximately

0.174 × 60 = 10.44 minutes,

so rounding to the nearest minute Tom spent about 4 hours and 10 minutes painting the fence.