Calculus ii chapter 6 section 4 volume using shells
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Calculus II Chapter 6 Section 4 Volume using Shells. David Dippel LoneStar College - Montgomery. Revolve the region bounded by y = x 2 +1 y = 0 and x = 0 around the y-axis. cross section. If we take a vertical slice. and revolve it about the y-axis. we get a cylinder.

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Calculus II Chapter 6 Section 4 Volume using Shells

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Calculus ii chapter 6 section 4 volume using shells

Calculus IIChapter 6 Section 4Volume using Shells

David Dippel

LoneStar College - Montgomery


Calculus ii chapter 6 section 4 volume using shells

Revolve the region bounded by y = x2+1 y = 0 and x = 0 around the y-axis.

cross section

If we take a vertical slice

and revolve it about the y-axis

we get a cylinder.

If we add all of the cylinders together, we can reconstruct the original object.


Calculus ii chapter 6 section 4 volume using shells

cross section

The volume of a thin, hollow cylinder is given by:

r is the x value of the function.

h is the y value of the function.

thickness is dx.


Calculus ii chapter 6 section 4 volume using shells

This is called the shell method because we use cylindrical shells.

cross section

If we add all the cylinders from the smallest to the largest:


Calculus ii chapter 6 section 4 volume using shells

Find the volume generated when this shape is revolved about the y axis.

We can’t solve for x, so we can’t use a horizontal slice directly.


Calculus ii chapter 6 section 4 volume using shells

If we take a vertical slice

and revolve it about the y-axis

we get a cylinder.

Shell method:


Calculus ii chapter 6 section 4 volume using shells

Note:

When entering this into the calculator, be sure to enter the multiplication symbol before the parenthesis.


Calculus ii chapter 6 section 4 volume using shells

When the strip is parallel to the axis of rotation, use the shells.

When the strip is perpendicular to the axis of rotation, use disks.

Long side of Rectangle

Short Side of Rectangle

Distance to Rectangle

Long side of Rectangle

Short Side of Rectangle

p


Find the volume of the region generated by rotation y 2x y 4 and x 0 around the x axis the line y 5

Find the volume of the region generated by rotation y = 2x, y = 4, and x = 0 around:the x-axisthe line y = 5


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