Chapter 6

Applications of the Definite Integral

6.1 Area Between Two Curves

Integrating with respect to x:

Top - Bottom dx( )y f x

Integrating with respect to y:

Right - Left dx( )x f y

Complete p 419 #7–25 Odd

6.2 Volumes by SlicingDisks and Washers

Rotational Volumes:

Disks

Washers

Area of a sliceV

2r2 2R r

Examples:

1. Find the volume of the solid that is obtained when the region under the curve

over the interval [1, 4] is revolved about the x–axis.

y x

Examples:

2. Find the volume of the solid generated when the region between the graphs of the equations

and over the interval [0, 2] is

revolved about the x–axis.

21( )

2f x x

( )g x x

Examples:

3. Find the volume of the solid generated when the region enclosed by , , and

is revolved about the y–axis.y x 2y

0x

Examples:

4. Find the volume of the solid generated when the region under the curve over the interval [0, 2] is rotated about the line y = –1.

2y x

Complete p 428 #11, 16, 20, 22, 23, 39, 40

6.3 Volumes by Cylindrical Shells

A cylindrical shell is a solid enclosed by two concentric right circular cylinders.

2 thicknessV r h

Volume by Cylindrical Shells about the y–axis:

Let f be continuous and nonnegative on [a, b] and let R be the region that is bounded above by y = f(x), below by the x–axis, and on the sides by x = a and x = b. Then the volume V of the solid of revolution that is generated by revolving the region R about the y–axis is given by

2 ( )b

aV x f x dx

Examples:

1. Use cylindrical shells to find the volume of the solid generated when the region R in the first quadrant enclosed between y = x and

y = x2 is revolved about the y – axis.

Examples:

2. Use cylindrical shells to find the volume of the solid generated when the region R under y = x2 over the interval [0, 2] is revolved about the line y = –1.

Complete p 436 #5–11 Odd, 13–16 All

6.4 Length of a Plane Curve

If y = f(x) is a smooth curve (f’ is continuous) on the interval [a, b], then the arc length L of this curve over [a, b] is defined as

21 ( )

b

aL f x dx

Examples:

1. Find the exact arc length of the curve over the interval from x = 0 to x = 1.

3

23 1y x

Examples:

2. Find the exact arc length of the curve over the interval from

y = 0 to y = 1.

3

2 21

23

x y

Complete p 441 #5–8, 22, 25

6.6 WorkLesson 1

If a constant force is applied:

Work = Force · Distance

Units:

Force Work

newtons (N) newton–meters (N–m)

joule (J), aka N–m

pounds (lb) foot–pounds (ft–lb)

If a variable force is applied:

Work =

where F(x) is the variable force and the force is applied over the interval [a, b].

( )b

aF x dx

Read “Hooke’s Law” on pg 452

Hooke’s Law

F(x) = kx where k is the spring constant.

Examples:

1. A spring has a natural length of 10 inches. An 800–lb force stretches the spring to 14 inches.

A. Find the spring constant.

B. How much work is done in stretching the spring from 10 inches to 12 inches?

C. How far beyond its natural length will a 1600–lb force stretch the spring?

Examples:

2. It takes a force of 21,714 lb to compress a coil spring assembly on a New York City transit Authority subway car from its free height of 8 inches to its fully compressed height of 5 inches.

A. What is the assembly’s spring constant?

B. How much work does it take to compress the assembly the first half inch? The second half inch? Answer to the nearest inch–pound.

Examples:

3. A bathroom scale is compressed inch when a 150–lb person stands on it. Assuming the scale behaves like a spring that obeys Hooke’s Law,

A. How much does someone who compresses the scale inch weigh?

B. How much work is done in compressing the scale inch?

1

16

1

8

1

8

Complete p 456 #6 – 9

6.6 Work Lesson 2

Work with Emptying and Filling:

Divide substance into layers and integrate the force required to move each layer to the top.

1. Find the area, A(y), of a slice.

2. Calculate the force required to move a slice to the top based on the position of the slice:

3. Integrate over the depth of the substance.

Depth of slice

0( ) ( ) Conversion constant F y A y dy

( ) ( ) Conversion constant Depth of substanceF y A y

( )F y dy

Examples:

17. The rectangular tank shown here, with its top at ground level, is used to catch runoff water. Assume that the water weighs 62.4 lb/ft3. How much work does it take to empty the tank by pumping the water back to ground level once the tank is full?

Examples:

18. A vertical right cylindrical tank measures 30 ft high and 20 ft in diameter. It is full of kerosene weighing 51.2 lb/ft3. How much work does it take to pump the kerosene to the level of the top of the tank?

Examples:

20. The truncated conical container shown here is full of strawberry milkshake that weighs (4/9) oz/in3. As you can see, the container is 7 in. deep, 2.5 in. across at the base, and 3.5 in. across at the top. The straw sticks up an inch above the top. About how much work does it take to drink the milkshake through the straw? Answer in inch–ounces.

Examples:24. Your town has decided to drill a well to

increase its water supply. As the town engineer, you have determined that a water tower will be necessary to provide the pressure needed for distribution, and you have designed the system shown here. The water is to be pumped from a 300–ft well through a vertical 4–in. pipe into the base of a cylindrical tank 20 ft in diameter and 25 ft high. The base of the tank will be 60 ft above the ground. The pump is a 3–hp pump, rated at 1650 ft–lb/sec. To the nearest hour, how long will it take to fill the tank the first time? (Include the time it takes to fill the pipe.) Assume water weighs 62.4 ft/lb3.

Complete p 457 #14–20

6.6 WorkLesson 3

Work with Leaky Containers:F(x) = Original weight · Percentage left at elevation y

Examples:

1. A leaky bucket weighs 22 newtons empty. It is lifted from the ground at a constant rate to a point 20 m above the ground by a rope weighing 0.4 N/m. The bucket starts with 70 N of water, but it leaks at a constant rate and just finishes draining as the bucket reaches the top. Find the amount of work done

A) lifting the bucket alone

B) lifting the water alone

C) lifting the rope alone

D) lifting the bucket, water, and rope together

Examples:

2. A bag of sand originally weighing 144 lb was lifted at a constant rate. The sand was half gone by the time the bag had been lifted 18 ft. How much work was done lifting the sand this far? (Neglect the weights of the bag and lifting equipment.)

Examples:

3. A 3 lb bucket containing 20 lb of water is hanging at the end of a 20 ft rope that weighs 4 oz/ft. The other end of the rope is attached to a pulley. How much work is required to wind the length of the rope onto the pulley, assuming that the rope is wound at a rate of 2 ft/sec and that as the bucket is being lifted, water leaks from the bucket at a rate of 0.5 lb/sec? (Calculate work with respect to time.)

Examples:

4. A rocket weighing 3 tons is filled with 40 tons of liquid fuel. In the initial part of the flight, fuel is burned off at a constant rate of 2 tons per 1000 ft of vertical height. How much work, in foot–tons, is done lifting the rocket 3000 feet?