Volumes by Disks and Washers

Or, how much toilet paper fits on one of those huge rolls, anyway??

Volume by SlicingApproximating area rectanglesVolume = length x width x height

Total volume = (A x t)

Volume of a slice = Area of a slice x Thickness of a slice A t

Volume by SlicingTotal volume = (A x t)

VOLUME = A dt

But as we let the slices get infinitely thin,

Volume = lim (A x t) t 0

Recall: A = area of a slice

Rotating a Function

Such a rotation traces out a solid shape (in this case, we get something like half an egg)

𝑦= 𝑓 (𝑥 )

Volume by Slices

Slice

r dt

Area of a slice

Disk Formula

VOLUME = A dt

VOLUME = , so…

distance between function and axis of rotation (axis – function)

and represent boundaries of the region (x – values if rotating about a horizontal axis)

Volume by Disks (Rotation about the x – axis)

r

thickness

x axis

y axis Slice

radius

𝑓 (𝑥)−0

𝑑𝑥Thus,

𝑦= 𝑓 (𝑥 )

VOLUME = but and , so...

𝑦=0

ExampleFind the volume of the solid generated by rotating the region bounded by about the x – axis.

radius =

Cone: V = =

Example Find the volume of the solid generated by rotating the region bounded by about the x – axis.

Homework: p. 324 #1-4*Sketch the region first!

ExampleRotate the region bounded by about the y – axis. Calculate the volume of the solid.

𝑦=√𝑥→𝑥=𝑦2

More Volumes

f(x)g(x)

rotate around x axis

SliceR

r

Area of a slice =

Area of big circle – hole

Washer Formula

VOLUME = A dt

VOLUME =

Volumes by Washers (about the x – axis)

f(x)g(x)

SliceR

r

dt

Big Rlittle r

𝑔 (𝑥 )−0

𝑓 (𝑥 )−0

Thus, dx

V =

2

The application we’ve been waiting for...

1

rotate around x axis1

0.5

f(x)

g(x)

Toilet Paperf(x)

g(x)12

0.51

So we see that:f(x) = 2, g(x) = 0.5

0 V = x only goes from 0 to 1,so we use these as the limits of integration. Now, plugging in our values for f and g: V =

Example Rotate the region enclosed by about the y – axis. Calculate the volume of the solid.Remember – radius = distance from function to axis!Big radius = Little radius =

𝑦=√𝑥→𝑥=𝑦2

Homework: p. 324 #5, 6, 8, 9*Sketch the region first!

Example Find the volume of the solid generated by rotating the region bounded by and about the x – axis. Big radius = Little radius =

Example Find the volume of the solid generated by rotating the region bounded by and about the line . Radius =

(horizontal)

By symmetry:

Volume by Cross SectionsRecall slices: VOLUME = A dt

A = area of cross sections of the figure need area of common shapes

Use x when cross-sections are perpendicular to the x – axis.

Use y when perpendicular to the y – axis.

Example Find the volume of a solid whose base is the region bounded by and whose cross sections perpendicular to the x – axis are equilateral triangles.Area of = Base = = Height of

base

base

A = (2−𝑥) ∙

Homework: handout #55, 56

TI-84 Calculator Methods Area between curves = To find intersection points: Type each curve into the screen

#5: intersect First curve? One equation should show in the

top left push Second curve? Other equation shows Guess? Estimate where it looks like they cross

type in your guess, Intersection point will be given

TI-84 Calculator Methods Zero of a function: #2: zero Area under ONE curve can be done by using #7:

To calculate any DEFINITE integral: #9: fnInt(

Type information for your problem fnInt(function, , lower bound, upper bound) Function = whatever you are integrating If calculating volume, remember to multiply

your answer by if necessary

Test – Area and Volume Area between curves Volume by:

– Disks and washers– Cross-sections

Integration on calculator

Review: p. 324 #7 (), 10 (), 11 (); p. 339 #26 ()