Nothing Special   »   [go: up one dir, main page]
Scribd Logo
Upload

Welcome to Scribd!

  • 12 views

    6.3 Lecture Notes

    Uploaded by

    Devil Inside
    The document discusses several methods for using integrals to calculate the volume of solid objects: 1) The slicing method, which involves slicing the solid into thin cross-sections and adding their volumes. 2) The disk/washer method for solids of revolution, which involves rotating an area about an axis and treating each slice as a disk/washer to calculate volume. 3) Examples are given of using these methods to calculate the volumes of various solids.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd

    6.3 Lecture Notes

    Uploaded by

    Devil Inside
    12 views5 pages
    The document discusses several methods for using integrals to calculate the volume of solid objects: 1) The slicing method, which involves slicing the solid into thin cross-sections and adding their volumes. 2) The disk/washer method for solids of revolution, which involves rotating an area about an axis and treating each slice as a disk/washer to calculate volume. 3) Examples are given of using these methods to calculate the volumes of various solids.

    Original Description:

    Original Title

    6.3-Lecture-Notes

    Copyright

    © © All Rights Reserved

    Available Formats

    PDF, TXT or read online from Scribd

    Did you find this document useful?

    Is this content inappropriate?

    The document discusses several methods for using integrals to calculate the volume of solid objects: 1) The slicing method, which involves slicing the solid into thin cross-sections and adding their volumes. 2) The disk/washer method for solids of revolution, which involves rotating an area about an axis and treating each slice as a disk/washer to calculate volume. 3) Examples are given of using these methods to calculate the volumes of various solids.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    12 views5 pages

    6.3 Lecture Notes

    Uploaded by

    Devil Inside
    The document discusses several methods for using integrals to calculate the volume of solid objects: 1) The slicing method, which involves slicing the solid into thin cross-sections and adding their volumes. 2) The disk/washer method for solids of revolution, which involves rotating an area about an axis and treating each slice as a disk/washer to calculate volume. 3) Examples are given of using these methods to calculate the volumes of various solids.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 5

    §6.

    3 Volume by Slicing, Page 1

    We have seen that integration is used to compute the area of two-dimensional regions bounded by
    curves. Integrals are also used to find the volume of three-dimensional regions (or solids). Once
    again, the slice-and-sum method is key to solving these problems.

    General Slicing Method


    Suppose a solid object extends from x = a to x = b, and the
    cross section of the solid perpendicular to the x-axis has an
    area given by the function A that is integrable on [a, b].
    Z b
    The volume of the solid is V = A(x)dx.
    a

    x
    1. Consider the solid S with a triangular base formed by y = , y = 0, and x = 4, for which
    2
    parallel cross-sections perpendicular to the base and x-axis are squares. Calculate the volume
    of S.

    2. Consider the solid S whose base is bounded by x = y 2 and x = 1, for which parallel
    cross-sections perpendicular to the base and x-axis are equilateral triangles. Calculate the
    volume of S.
    §6.3 Volume by Slicing, Page 2

    Solids of Revolution - Disk Method about the x-axis


    Suppose you are given a region R in the xy-plane and you rotate R about one of the axes. This
    results in a 3-dimensional solid, S. We can use the following approach to calculate the volume of S.
    (1) “Fill up” the region R with rectangles, each with width ∆x and height f (x).
    (2) Rotate each rectangle around the x-axis to form a collection of circular disks.
    (3) Calculate the volume of each circular disk.
    (4) Sum up the volumes of each disk to approximate the total volume of the solid S.

    Let f be continuous with f (x) ≥ 0 on the interval [a, b]. If


    the region R bounded by the graph of f , the x-axis, and the
    lines x = a and x = b is revolved about the x-axis, the volume
    Z b
    of the resulting solid of revolution is V = π(f (x))2 dx,
    a
    where f (x) describes the disk radius.

    √
    3. Consider the region bounded by y = x, the x-axis, and the line x = 1.
    Calculate the volume of the solid generated by a rotation about the x-axis.
    §6.3 Volume by Slicing, Page 3

    Solids of Revolution - Washer Method about the x-Axis


    A slight variation on the disk method enables us to compute the volume of more exotic solids of
    revolutions. Let f and g be continuous functions with f (x) ≥ g(x) ≥ 0 on the interval [a, b].
    Let R be the region bounded by y = f (x), y = g(x), and the
    lines x = a and x = b. When R is revolved about the x-axis, the
    volume of the resulting solid of revolution is
    Z b
    V = π(f (x)2 − g(x)2 )dx,
    a
    where f (x) describes the outer radius and g(x) describes the inner
    radius.

    4. Consider the region bounded by y = x2 + 2, y = 21 x + 1, x = 0, and x = 1.


    Calculate the volume of the solid generated by a rotation about the x-axis.
    §6.3 Volume by Slicing, Page 4

    Solids of Revolution about the y-Axis


    5. Consider the region bounded by y = x3 , y = 1, y = 8, and x = 0. Calculate the volume of the
    solid generated by a rotation about the y-axis.

    Solids of Revolution about Other Lines


    6. Consider the region bounded by y = 0, x = 0, and y = 1 − x. Calculate the volume of the
    solid generated by a rotation about the line x = 2.
    §6.3 Volume by Slicing, Page 5

    Additional Practice Problems


    7. Find the volume of the torus (doughnut) formed when the circle of radius 2 centered at (3, 0)
    is revolved about the y-axis. Use symmetry and geometry to evaluate this integral.

    8. Use calculus to find the volume of a right circular cone with height h and base radius r.

    9. A 6 inch tall plastic cup is shaped like a surface obtained by rotating a line segment in the
    first quadrant about the x-axis. Given that the radius of the base of the cup is 1 inch, the
    radius of the top of the cup is 2 inches, and the cup is filled to the brim with water, use
    integration to approximate the volume of water in the cup.

    You might also like