7.3

Volumes

Quick Review

Give a formula for the area of the plane region in terms

of the single variable .

1. a square with side length .

2. a semicircle of radius .

3. a semicircle of diameter .

4. an equilateral triangle

x

x

x

x

with sides of length .

5. an isosceles triangle with two sides of length 2

and one of length .

x

x

x

2x

2/ 2x8/ 2x

2 4/3 x

2 4/15 x

What you’ll learn about Volumes As an Integral Square Cross Sections Circular Cross Sections Cylindrical Shells Other Cross Sections

Essential QuestionHow can we use calculus to computevolumes of certain solids in three dimensions?http://www.math.psu.edu/dlittle/java/calculus/volumewashers.html

http://www.math.psu.edu/dlittle/java/calculus/volumedisks.html

xy 3 .1

Find the volume of the solid when the curve is rotated around the x-axis.

11 x

1 .2 2 xy

Find the volume of the solid when the curve is rotated around the x-axis.

22 x

3 .3 xy

Find the volume of the solid when the curve is rotated around the x-axis.

22 x

xy sin3 .4

Find the volume of the solid when the curve is rotated around the x-axis.

x0

xy cos2

1 .5

Find the volume of the solid when the curve is rotated around the x-axis.

22

x

xxy 2 .6 3

Find the volume of the solid when the curve is rotated around the x-axis.

30 x

xey2

1 .7

Find the volume of the solid when the curve is rotated around the x-axis.

20 x

1 .8 xy

Find the volume of the solid when the curve is rotated around the x-axis.

90 x

Volume of a Solid

b

adxxAV

The definition of a solid of unknown integrable cross section area A(x) from x = a to x = b is the integral of A from a to b,

How to Find Volumes by the Method of Slicing1. Sketch the solid and a typical cross section.

2. Find a formula for A(x).

3. Find the limits of integration.

4. Integrate A(x) to find the volume.

Example Square Cross Sections

1. A pyramid 3 m high has congruent triangular sides and a square base that is 3 m on each side. Each cross section of the pyramid parallel to the base is a square. Find the volume of the pyramid.

1. Sketch: Draw the pyramid with its vertex at the origin and its altitude along the interval 0 < x < 3.

Sketch a typical cross section at a point x between 0 and 3.

2. Find a formula for A(x):

The cross section at x is a square x meters on a side, so the formula will be:

2xxA

Example Square Cross Sections

1. A pyramid 3 m high has congruent triangular sides and a square base that is 3 m on each side. Each cross section of the pyramid parallel to the base is a square. Find the volume of the pyramid.

3. Find the limits of integration:

The square goes from x = 0 to x = 3.

4. Integrate to find the volume:

3

0 dxxAV

3

0

2 dxx3

0

3

3

1

x 9 m3

Example A Solid of Revolution

2. The region between the graph f (x) = 2 + x cos x and the x-axis over the interval [– 2, 2] is revolved about the x-axis to generate a solid. Find the volume of the solid.

http://www.math.psu.edu/dlittle/java/calculus/volumewashers.html

Revolving the region about the x-axis generates a ____________ solid.vase-shapedThe cross section at a typical point x is __________.circularThe radius is equal to ______. f (x)

xA 2xf

V 2

2 2cos2 xx dx

NINT ,cos2 2xx ,x ,2 2

43.52 3units

Example Finding Volumes Using Cylindrical Shells3. Find the volume of the solid generated by revolving about the x-axis the

region bounded by .3 and 12 xyxy

http://www.math.psu.edu/dlittle/java/calculus/volumewashers.html

xA 2 R 2 r 22 rR

Rr

V 2

1 23x 22 1 x dx

2

1V 4x 2x x6 8 dx

NINT ,8624 xxx ,x ,1 2

51.73 3units

Pg. 406, 7.3 #1-25 odd

Cylindrical Shell MethodUse the shell method when the axis of revolution is perpendicular to the axis containing the natural interval of integration.

Instead of summing volumes of thin slices, we sum volumes of thin cylindrical shells that grow outward from the axis of revolution.

4. The region bounded by the curve y = , the x-axis, and the line x = 4 is revolved about the x-axis to generate a solid. Find the volume of the solid.

x

b

adxxAV

hrxA 2 where

2

0 2 y 24 y dy 4 3yy

2 2 2y 4

4

1y 2

0

2 8 4 8 3units

Example Finding Volumes Using Cylindrical Shells5. The region bounded by the curve y = , y = x, and x = 0 is revolved

about the y-axis to form a solid. Use cylindrical shells to find the volume of the solid.

4 2x

xr

xxh 24

xx 24042 xx

562.1x

.5621

0 2 x xx 24 dx 4 23 xxx

NINT 2 ,4 23 xxx ,x ,0 562.1

327.13 3units

Example Other Cross Sections6. A solid is made so that its base is the shape of the region between the x-axis and one

arch of the curve y = 2 sin x. Each cross section cut perpendicular to the x-axis is a semicircle whose diameter runs from the x-axis to the curve. Find the volume of the solid.

b

adxxAV

2

2

1 where rxA

Radius of the semicircle is xf2

1

xsinV

0

2

1 2sin x dx

NINT 2

, sin 2x ,x ,0

47.2 3units

Pg. 251, 4.6 #1-35 odd

Quick Quiz Sections 7.1-7.3

-1

You may use a graphing calculator to solve the following problems.

1. The base of a solid is the region in the first quadrant bounded by

the -axis, the graph of sin , and the vertical line 1. x y x x For this

solid, each cross section perpendicular to the -axis is a square.

What is its volume?

(A) 0.117

(B) 0.285

(C) 0.467

(D) 0.571

(E) 1.571

x

Quick Quiz Sections 7.1-7.3

-1

You may use a graphing calculator to solve the following problems.

1. The base of a solid is the region in the first quadrant bounded by

the -axis, the graph of sin , and the vertical line 1. x y x x For this

solid, each cross section perpendicular to the -axis is a square.

What is its volume?

(A) 0.117

(B) 0.285

(D) 0.571

(C) 0.46

(E)

7

1.571

x

Quick Quiz Sections 7.1-7.3

2

2. Let be the region in the first quadrant bounded by the

graph of 3 - and the -axis. A solid is generated when

is revolved about the vertical line -1. Set up, but do not

integrate, the def

R

y x x x

R x

3 2

0

3 2

1

3 2

0

3 2

0

3 2

0

inite integral that gives the volume of this solid.

(A) 2 1 3

(B) 2 1 3

(C) 2 3

(D) 2 3

(E) 3

x x x dx

x x x dx

x x x dx

x x dx

x x dx

Quick Quiz Sections 7.1-7.3

2

2. Let be the region in the first quadrant bounded by the

graph of 3 - and the -axis. A solid is generated when

is revolved about the vertical line -1. Set up, but do not

integrate, the def

R

y x x x

R x

3

3 2

1

3 2

0

3 2

0

3

2

0

2

0

inite integral that gives the volume of this solid.

(B) 2 1 3

(C) 2 3

(D) 2 3

(E) 3

(A) 2 1 3x x x

x x x dx

x x x d

d

x

x x dx

x x dx

x

Quick Quiz Sections 7.1-7.3

0.2

3. A developing country consumes oil at a rate given by

( ) 20 million barrels per year, where is time measured

in years, for 0 10. Which of the following expressions

gives the amount of oi

tr t e t

t

10

0

10

0

l consumed by the country during

the time interval 0 10?

(A) (10)

(B) (10) - (0)

(C) '( )

(D) ( )

(E) 10 (10)

t

r

r r

r t dt

r t dt

r

Quick Quiz Sections 7.1-7.3

0.2

3. A developing country consumes oil at a rate given by

( ) 20 million barrels per year, where is time measured

in years, for 0 10. Which of the following expressions

gives the amount of oi

tr t e t

t

10

0

10

0

l consumed by the country during

the time interval 0 10?

(A) (10)

(B) (10) - (0)

(C) '( )

(E)

(D)

1 (10)

(

0

)

t

r

r r

r t dt

r

r t dt