VolumeSection 7.3a

Recall a problem we did way back in Section 5.1…

Estimate the volume of a solid sphere of radius 4.

216f x x Each slice can be approximatedby a cylinder:

2V r h

2

2 216 1 16x x

Radius:Height: 216 x1

Volume of each cylinder:

By letting the height of each cylinder approach zero, we couldfind the exact volume using a definite integral!!!

Volume as an IntegralNow, we will use similar techniques to calculate volumes of manydifferent types of solids Let’s talk through Figure 7.16 on p.383

The volume of this cylinder is given by

kV base area x height kA x x And the following sum approximates the volume of theentire solid: k kV A x x

This is a Riemann sum for A(x) on [a, b]. We get betterapproximations as the partitions get smaller Their limitingintegral can be defined as the volume of the solid.

Definition: Volume of a Solid

The volume of a solid of known integrable cross section areaA(x) from x = a to x = b is the integral of A from a to b,

b

aV A x dx

How to Find Volume by the Method of Slicing

1. 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.

A Note: Cavalieri’s Theorem

If two plane regions can be arranged to lie over the same intervalof the x-axis in such a way that they have identical vertical crosssections at every point, then the regions have the same area.

a x b

Cross sections havethe same length atevery point in [a, b]

So these blue shaded regions have the exact same area!!!

This idea can be extended tovolume as well……take alook at Figure 7.17 on p.384.

Our First Practice Problem

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

Let’s follow our four-step process:

1. Sketch. Draw the pyramid with its vertex at the origin and itsaltitude along the interval . Sketch a typical crosssection at a point x between 0 and 3.

0 3x

2. Find a formula for A(x). The cross section at x is a square xmeters on a side, so

2A x x

Our First Practice Problem

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

Let’s follow our four-step process:

3. Find the limits of integration. The squares go from x = 0to x = 3.

4. Integrate to find the volume.

3

0V A x dx

3 2

0x dx

33

03

x

9 3m

Guided Practice

The solid lies between planes perpendicular to the x-axis at x = –1and x = 1. The cross sections perpendicular to the x-axis arecircular discs whose diameters run from the parabolato the parabola .

2y x22y x

Width of each cross section: 2 22w x x 22 2x

Area of each cross section: 2A x r2

2

w 221 x

How about a diagram of this solid?

Guided Practice

The solid lies between planes perpendicular to the x-axis at x = –1and x = 1. The cross sections perpendicular to the x-axis arecircular discs whose diameters run from the parabolato the parabola .

2y x22y x

To find volume, integrate these areas with respect to x:

1 22

11V x dx

1 4 2

12 1x x dx

1

5 3

1

1 2

5 3x x x

16

15

Guided PracticeThe solid lies between planes perpendicular to the x-axis at x = –1and x = 1. The cross sections perpendicular to the x-axis betweenthese planes are squares whose diagonals run from the semi-

21y x 21y x circle to the semicircle .

How about a diagram of this solid?

22 1w x Cross section width:

2A x sCross section area:

2

2

w

222 1

2

x

22 1 x

Guided PracticeThe solid lies between planes perpendicular to the x-axis at x = –1and x = 1. The cross sections perpendicular to the x-axis betweenthese planes are squares whose diagonals run from the semi-

21y x 21y x circle to the semicircle .

1 2

12 1V x dx

Volume:

13

1

12

3x x

2 2

23 3

8

3

Guided PracticeThe solid lies between planes perpendicular to the x-axis at and . The cross sections perpendicular tothe x-axis are circular discs with diameters running from the curve to the curve .

3x 3x

tany x secy xThe diagram?

sec tanw x x Cross section width:

2A x rCross section area:

2

2

w

2sec tan4

x x

Guided PracticeThe solid lies between planes perpendicular to the x-axis at and . The cross sections perpendicular tothe x-axis are circular discs with diameters running from the curve to the curve .

3x 3x

tany x secy x

3 2

3sec tan

4V x x dx

Volume:

3 2 2

3sec 2sec tan tan

4x x x x dx

3

3tan 2sec tan

4x x x x

3 2 2

3sec 2sec tan sec 1

4x x x x dx

Guided PracticeThe solid lies between planes perpendicular to the x-axis at and . The cross sections perpendicular tothe x-axis are circular discs with diameters running from the curve to the curve .

3x 3x

tany x secy x

3

3

1tan sec

2 2x x x

3

3tan 2sec tan

4x x x x

2

36

3 2 3 22 6 6