Volume

Lesson #3 of

Unit 6. Applications of Integration

(Textbook 6.2)

Approximating Volume

Volume of any cylindrical figure is , that is the product of its bottom area and height.

Now suppose we have a solid S, as shown below. To approximate its volume let’s divide S into n “slabs” of equal width x. (Think of slicing a loaf of bread.) Then approximate volume of each slab is the product of cross-section area and x, which is the height of slabs.

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<A cross section of solid> <Approximated volume of solid by slabs>

Approximation to Actual Volume

Continuing from the previous slide, the approximation of volume can be generalized as following.

, where is a sample point

This approximation becomes the actual volume as . Thus we obtain following.

When we use the volume formula, it is important to remember that A(x) is the area of a moving cross-section obtained by slicing through x perpendicular to the x-axis.

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Example

Find the volume of the solid obtained by rotating about the x-axis the region under the curve from 0 to 1.

Find the volume of the solid obtained by rotating the region bounded by ,

, and about the -axis.

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Practice

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

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about the x-axis

about the x-axis about the y-axis

Solids of Revolution

Problems from the previous slides are examples of solids of revolution because they are obtained by revolving a region about a line. In general, we calculate the volume of a solid of revolution by using one of following formulas, depending on the axis of revolution.

The idea is that f(x) or f(y) is serve as the radius of disc whose area is

.

When a solid of revolution is created by rotating an area between curves, above formulas are modified to following.

and

…where f(x) or f(y) serve as the outer radius and g(x) or g(y) as the inner.

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<Volume when rotated about x-axis> <Volume when rotated about y-axis>

Example

The region enclosed by the curves and is rotated about the x-axis. Find the volume of the resulting solid.

Find the volume of the solid obtained by rotating the region in the above problem about the line y = 2.

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Practice

Find the volume of the solid obtained by rotating the region in the previous problem about the line x = -1.

Find the volume of the solid obtained by rotating the region bounded by

and about the x-axis.

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Practice

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

about the y-axis about y = 2

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Solids of Known Cross-Sections

Solids of revolution have disc or washer shape cross-sections, so we used the circle area formula to set-up the definite integral that enabled us to calculate the volume. Not all solids are solids of revolution, but as long as we can formulate a formula for the cross-section of given solid, we can calculate its volume by the general formula for the volume of solid, .

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Figure shows a solid with a circular base of radius 1. Parallel cross-sections perpendicular to the base are equilateral triangles. Find the volume of the solid (Area of Equilateral Triangle: , where s is the side length).

Example

Show that the volume of a sphere of radius r is

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Practice

Find the volume of a pyramid whose base is a square with side L and whose height is h.

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Suggested Problems

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1, 3, 5, 7, 9, 11, 47, 48, 50

Suggested Problems - SOLUTIONS -

Textbook p.438 – 439

1) 3) 5)

7) 9)

11)

47)

Suggested Problems - SOLUTIONS -

Textbook p.438 – 439

48) 50)