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The region enclosed by the The region enclosed by the

The region enclosed by the - PowerPoint Presentation

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The region enclosed by the - PPT Presentation

x axis and the parabola is revolved about the line x 1 to generate the shape of a cake What is the volume of the cake DO NOW The Shell Method Section 73d The region enclosed by the ID: 500777

shell volume solid axis volume shell axis solid region let

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Slide1

The region enclosed by the x-axis and the parabola

is revolved about the line

x

= –1 to

generate the shape of a cake. What is the volume of the cake?

DO NOWSlide2

The Shell Method

Section 7.3dSlide3

The region enclosed by the

x

-axis and the parabola

is revolved about the line

x

= –1 to

generate the shape of a cake. What is the volume of the cake?

x

= –1

Let’s first try to integrate with

respect to

y

 Washers!!!

To find the inner and outer radii of our washer, we would

need

to get the equation of the original parabola in terms of

x

...

n

ot an easy task…Slide4

The region enclosed by the x-axis and the parabola

is revolved about the line

x

= –1 to

generate the shape of a cake. What is the volume of the cake?

Instead of slicing horizontally,

we will cut a series of

cylindrical

s

lices by cutting straight down

all the way around the solid.

The radii of the cylinders gradually increase, and the heights of

t

he cylinders follow the contour of the parabola: smaller to larger,

t

hen back to smaller.

When we unroll one of these

shells

, we have a rectangular

p

rism, with height equal to the height of the shell, length

e

qual to the circumference of the shell, and thickness equal

t

o .Slide5

The region enclosed by the x-axis and the parabola

is revolved about the line

x

= –1 to

generate the shape of a cake. What is the volume of the cake?

Volume of each shell:

Here, we have

To find the total volume of the solid, integrate with respect to

x

:Slide6

The region bounded by the curve , the x-axis, and the

line x = 4 is revolved about the

x-axis to generate a solid. Findt

he volume of the solid.

Let’s solve this problem first by integrating with respect to

x

:

Cross section area:

Volume:Slide7

The region bounded by the curve , the x-axis, and the

line x = 4 is revolved about the

x-axis to generate a solid. Findt

he volume of the solid.

Now let’s integrate with respect to

y

using our new technique:

Radius of each shell:

Height of each shell:

Limits of integration:

0 to 2

Thickness of each shell:Slide8

The region bounded by the curve , the x-axis, and the

line x = 4 is revolved about the

x-axis to generate a solid. Findt

he volume of the solid.

Now let’s integrate with respect to

y

using our new technique:

Volume:Slide9

Let’s redo a problem from a previous class, this time using shells…

Find the volume of the solid generated by revolving the region

bounded above by the square root function and below by the

identity function about the

y

-axis

Cross section area:

Volume:

Our previous solutionSlide10

Let’s redo a problem from a previous class, this time using shells…

Find the volume of the solid generated by revolving the region

bounded above by the square root function and below by theidentity function about the

y-axisShell radius:

Volume:

Shell height:

Shell thickness:Slide11

Let’s redo a problem from a previous class, this time using shells…

Find the volume of the solid generated by revolving the triangularregion bounded by the lines y

= 2x,

y = 0, and x = 1 about thel

ine x = 2.

R

r

Cross section area:

Volume:

Our previous solutionSlide12

Let’s redo a problem from a previous class, this time using shells…

Find the volume of the solid generated by revolving the triangularregion bounded by the lines y

= 2x,

y = 0, and x = 1 about thel

ine x = 2.

Shell radius:

Shell height:

Shell thickness:

Volume: