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VOLUME OF A SOLID WITH KNOWN CROSS SECTION

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Introduction

volumeofasolid1
We have already learned to find the area of a plane region bounded by two curves which is obtained by integrating the length of a general cross section over an appropriate interval.  Here we will see that the same basic principle can be used to find volumes of certain three dimensional solids.


Let S be a solid that extends along the x-axis and is bounded on the left and right, respectively, by the planes that are perpendicular to the x-axis at x=a and x=b.  We are finding the volume V of the solid, assuming that its cross-sectional area A(x) is known at each x in the interval [a, b].

To solve this problem we divide the interval [a, b] into n subintervals, which has the effect of dividing the solid into n slabs [fig (ii)]

volumeofasolid2

If we assume that the width of the kth slab is ∆xk, then the volume of the slab can be approximated by the volume of a right cylinder of width (height) ∆xk and cross-sectional area A(xk*), where xk* is a number in the kth subinterval.  Adding these approximations yields the following Riemann sum that approximates the volume V:
                                   V ≈ ΣA(xk*)∆xk
Taking the limit as n increases and the widths of the subintervals approach zero yields the definite integral

  1.     V     =     lim     ΣA(xk*)∆xk     =     ab A(x)dx
                       max∆xk   0
    We can conclude the result in the following way,
    Volume formula
    Let S be a solid bounded by two parallel planes perpendicular to the x-axis at x=a and x=b.  If, for each x in [a, b] the cross-sectional area of S perpendicular to the x-axis is A(x), then the volume of the solid is,
                                   V= ab A(x) dx provided A(x) is integrable.
    Volume Formula
    Let S be a solid bounded by two parallel planes perpendicular to the y-axis at y=c and y=d.  If, for each y in [c, d], the cross-sectional area of S perpendicular to the y-axis is A(y), then the volume of the solid is,
                                  V = cd A(y) dy provided A(y) is integrable.
    In words, these formulas states that, “The volume of a solid can be obtained by integrating the cross-sectional area from one end of the solid to the other”.
    Example: Find the formula for the volume of a right pyramid whose altitude is h and whose base is a square with sides of length a.
    Solution: We introduce a rectangular coordinate system in which the y-axis passes through apex and is perpendicular to the base, and the x-axis passes through the base and is parallel to a side of the base.
    At any ‘y’ in the interval [0, h] on the y-axis, the cross section perpendicular to the y-axis is a square.  If ‘s’ denotes the length of a side of this square, then by similar triangles.

volumeofasolid3

Solids of revolution

A solid of revolution is a solid that is generated by revolving a plane region about a line that lies in the same plane as the region; the line is called the axis of revolution.

volumeofasolid4

Volume of a solid of revolution

Let f be continuous and non-negative on [a, b] and let R be the region that is bounded by y=f(x), below by the x-axis, and on the sides by the lines x=a and x=b, then the volume of the solid or revolution that is generated by revolving the region R about the x-axis is given by
                                    V= ab π[f(x)]2dx
                                      = ab π y2 dx
                                       = π aby2 dx                              

                                  volumeofasolid5


Example: Find the volume of a paraboloid of revolution formed by revolving the parabola y2=8x about the x-axis from x=0 to x=6
Solution: The equation of the parabola is y2=8x,
Hence volume = 06 πy2dx
                     = π06 8x dx
                     = 8π[x2/2]06
                     = 8π x ½ [62-02]
                     =4πx36
                     = 144πcubic units.

Volume by cylindrical shells


A cylindrical shell is a solid enclosed by two concentric right circular cylinders.  The volume V of a cylindrical shell with inner radius r1, outer radius r2, and height h can be written as
                         V = (area of cross section).height
Let f be continuous and non-negative on [a, b] and let R be the region that is bounded above by y=f(x) below by the x-axis, and on the sides by the lines x=a and x=b.  Then the volume V of the solid revolution that is generated by revolving the region R about the y-axis is given by
                       V = ab 2πxf(x)dx

  1.                           volumeofasolid6
      
    Example: Use cylindrical shells to find the volume of the solid generated when the region R in the first quadrant enclosed between y=x and y=x2 is revolved about the y-axis.
    Solution: V= 012πx(x-x2)dx = 2π 01 (x2-x3)dx
                                     = 2π [(1/3) – (1/4)]
                                     = π/6 cubic units.

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Reference Links:

http://www.intmath.com/applications-integration/2-area-under-curve.php

http://www.cliffsnotes.com/study_guide/Volumes-of-Solids-with-Known-Cross-Sections.topicArticleId-39909,articleId-39906.html

http://en.wikipedia.org/wiki/Solid_of_revolution

http://en.wikipedia.org/wiki/Shell_integration

http://en.wikipedia.org/wiki/Riemann_sum

    

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