Volumes

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7.3

• Volumes

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Quick Review

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What youll learn about

• Volumes As an Integral
• Square Cross Sections
• Circular Cross Sections
• Cylindrical Shells
• Other Cross Sections
• Essential Question
• How can we use calculus to compute
• volumes of certain solids in three dimensions?

http//www.math.psu.edu/dlittle/java/calculus/volu
mewashers.html
http//www.math.psu.edu/dlittle/java/calculus/volu
medisks.html
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Find the volume of the solid when the curve is
rotated around the x-axis.
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Find the volume of the solid when the curve is
rotated around the x-axis.
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Find the volume of the solid when the curve is
rotated around the x-axis.
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Find the volume of the solid when the curve is
rotated around the x-axis.
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Find the volume of the solid when the curve is
rotated around the x-axis.
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Find the volume of the solid when the curve is
rotated around the x-axis.
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Find the volume of the solid when the curve is
rotated around the x-axis.
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Find the volume of the solid when the curve is
rotated around the x-axis.
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Volume of a Solid
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 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.

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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.

Draw the pyramid with its vertex at the origin
and its altitude along the interval 0 lt x lt 3.
1. Sketch
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
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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
m3
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Example A Solid of Revolution

1. 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.

vase-shaped
Revolving the region about the x-axis generates a
____________ solid.
circular
The cross section at a typical point x is
__________.
f (x)
The radius is equal to ______.
http//www.math.psu.edu/dlittle/java/calculus/volu
mewashers.html
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Example Finding Volumes Using Cylindrical Shells
http//www.math.psu.edu/dlittle/java/calculus/volu
mewashers.html
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Pg. 406, 7.3 1-25 odd
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Cylindrical Shell Method
Use 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.
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Example Finding Volumes Using Cylindrical Shells
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Example Other Cross Sections
Radius of the semicircle is
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Pg. 251, 4.6 1-35 odd
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Quick Quiz Sections 7.1-7.3

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Quick Quiz Sections 7.1-7.3

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Quick Quiz Sections 7.1-7.3

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Quick Quiz Sections 7.1-7.3

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Quick Quiz Sections 7.1-7.3

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Quick Quiz Sections 7.1-7.3