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

Welcome to Scribd!

  • 234 views

    Volume Integrals

    Uploaded by

    Karthikeya
    The document discusses volume integrals. It defines a volume integral as the integral of a scalar or vector function over a volume V enclosed by a surface S in the xyz plane. For a scalar function f, the volume integral is defined as the triple integral of f over the bounds of V. For a vector function F, the volume integral is defined as the triple integral of F over the bounds of V. An example problem is then worked out to evaluate the volume integral of the divergence of a vector function F over a given volume V bounded by several planes.

    Copyright:

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

    Volume Integrals

    Uploaded by

    Karthikeya
    234 views1 page
    The document discusses volume integrals. It defines a volume integral as the integral of a scalar or vector function over a volume V enclosed by a surface S in the xyz plane. For a scalar function f, the volume integral is defined as the triple integral of f over the bounds of V. For a vector function F, the volume integral is defined as the triple integral of F over the bounds of V. An example problem is then worked out to evaluate the volume integral of the divergence of a vector function F over a given volume V bounded by several planes.

    Original Title

    5._Volume_Integrals

    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 volume integrals. It defines a volume integral as the integral of a scalar or vector function over a volume V enclosed by a surface S in the xyz plane. For a scalar function f, the volume integral is defined as the triple integral of f over the bounds of V. For a vector function F, the volume integral is defined as the triple integral of F over the bounds of V. An example problem is then worked out to evaluate the volume integral of the divergence of a vector function F over a given volume V bounded by several planes.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    234 views1 page

    Volume Integrals

    Uploaded by

    Karthikeya
    The document discusses volume integrals. It defines a volume integral as the integral of a scalar or vector function over a volume V enclosed by a surface S in the xyz plane. For a scalar function f, the volume integral is defined as the triple integral of f over the bounds of V. For a vector function F, the volume integral is defined as the triple integral of F over the bounds of V. An example problem is then worked out to evaluate the volume integral of the divergence of a vector function F over a given volume V bounded by several planes.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    You are on page 1of 1

    VOLUME INTEGRALS:

    Let f be a scalar function and V is a volume enclosed by a surface S in xyz plane then the
    integral of scalar function f over volume V is volume integral and is given by
    x2 y 2 z 2

    V=  V
    f dxdydz     f dzdydx
    x1 y1 z1

    (OR)
    Let F be a vector function and V is a volume enclosed by a surface S in xyz plane then the
    integral of vector function F over volume V is volume integral and is given by
    x2 y 2 z 2

    V= V
    F dxdydz     F dzdydx
    x1 y1 z1

    PROBLEMS:

    1. If F  (2 x 2  3z)i  2xyj  4 xk Evaluate    Fdv


    V
    where V is the closed region

    bounded by the planes x = 0, y = 0, z = 0 and the plane 2x+2y+z = 4


       
    Sol:   F   i  j  k   (2 x 2  3z )i  2 xyj  4 xk = 4x-2x+0 = 2x.
     x y z 

    Limits of z: 0 to 4-2x-2y

    Limits of y: 0 to 2-x

    Limits of x: 0 to 2
    2 2 x 4 2 x 2 y 2 2 x 4 2 x  2 y 2 2 x

       Fdv  
    V
    0

    0
     2 x dzdydx  
    0 0
     2 x z 
    0
    dydx  
    0
     2 x 4  2 x  2 ydydx
    0
    0

    2 2 x 2 2 x 2 x
     2
    y2 
     4
    0
     x 2  x  ydydx  4
    0 0
    0 x ( 2  x )  xy dydx  4 0  x ( 2  x ) y  x  dx
    2 0
    2
     (2  x) 2 
    2 2
     4   x( 2  x) 2  x dx  2  x ( 2  x ) 2
    dx  2  x(4  x 2  4 x)dx
    0 
    2  0 0

    2
    2
     x4 x3   32   32  8
     2 (4 x  x  4 x )dx  22 x 2 
    3 2
     4   28  4    212   
    0  4 3 0  3  3 3

    22
    Dr. K.S. Balamurugan, RVR & JC College of Engineering, Guntur, A.P

    You might also like