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Volumes by Slicing Help (page 2)

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By — McGraw-Hill Professional
Updated on Sep 1, 2011

Volume of a Solid with a Square Cross-Section Example

A solid has base in the x-y plane consisting of a unit square with center at the origin and vertices on the axes. The vertical cross-section at position x is itself a square. Refer to Fig. 8.9. What is the volume of this solid?

Applications of the Integral 8.1 Volumes by Slicing

Fig. 8.9

Solution

It is sufficient to calculate the volume of the right half of this solid, and to double the answer. Of course the extent of x is then Applications of the Integral 8.1 Volumes by Slicing. At position x, the height of the upper edge of the square base is Applications of the Integral 8.1 Volumes by Slicing. So the base of the vertical square slice is Applications of the Integral 8.1 Volumes by Slicing ( Fig. 8.10 ). The area of the slice is then

Applications of the Integral 8.1 Volumes by Slicing

Applications of the Integral 8.1 Volumes by Slicing

Fig. 8.10

It follows that

Applications of the Integral 8.1 Volumes by Slicing

     Applications of the Integral 8.1 Volumes by Slicing

You Try It : Calculate the volume of the solid with base in the plane an equilateral triangle of side 1, with base on the x -axis, and with vertical cross-section parallel to the y -axis consisting of an equilateral triangle.

Volume Inside a Sphere with a Disk Cross-Section Example

Calculate the volume inside a sphere of radius 1.

Solution

It is convenient for us to think of the sphere as centered at the origin in the x-y plane. Thus ( Fig. 8.11 ) the slice at position x, −1 ≤ x ≤ 1, is a disk. Since we are working with base the unit circle, we may calculate (just as in Example 8.2) that the diameter of this disk is Applications of the Integral 8.1 Volumes by Slicing. Thus the radius is Applications of the Integral 8.1 Volumes by Slicing and the area is

Applications of the Integral 8.1 Volumes by Slicing

Applications of the Integral 8.1 Volumes by Slicing

Fig. 8.11

In conclusion, the volume we seek is

Applications of the Integral 8.1 Volumes by Slicing

We easily evaluate this integral as follows:

Applications of the Integral 8.1 Volumes by Slicing

You Try It: Any book of tables (see [CRC]) will tell you that the volume inside a sphere of radius r is 4 πr 3 /3. This formula is consistent with the answer we obtained in the last example for r = 1. Use the method of this section to derive this more general formula for arbitrary r.

Find practice problems and solutions for these concepts at: Applications of the Integral Practice Test.

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