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Q3E

Expert-verifiedFound in: Page 816

Book edition
2nd

Author(s)
James Stewart

Pages
830 pages

ISBN
9781133112280

**Find the value of** **\(\iint_{H}{f}(x,y,z)dS \)**..

The value of **\(\iint_{H}{f}(x,y,z)dS \)** is 2827.

**The surface integral is a generalization of multiple integrals that allows for surface integration. The surface integral is sometimes referred to as the double integral. We can integrate across a surface in either the scalar or vector fields for any given surface. The function returns the scalar value in the scalar field and function returns the vector value in the vector field.**

**“The surface integral of f **

As given data;

\(H\) be the hemisphere \({x^2} + {y^2} + {z^2} = 50,z \ge 0\).

\(f(3,4,5) = 7,f(3, - 4,5) = 8,f( - 3,4,5) = 9\) and \(f( - 3, - 4,5) = 12\).

The \(x,z\)\( - and\)\(y,z\)\( - planesareusedtodividehemisphere\)(H) into four patches of equal size, each has a surface area equal to \(\frac{1}{8}\) the surface area of a sphere with radius \(\sqrt {50} \)Therefore;

\(\Delta S = \frac{1}{8}(4)\pi {(\sqrt {50} )^2} = \frac{1}{2}\pi (50) = 25\pi \)

The sample points in the four patches are\(( \pm 3, \pm 4,5)\).

\(\iint_H f(x,y,z)dS \cong \left\{ {\begin{aligned}{*{20}{l}}{[f(3,4,5)](25\pi ) + [f(3, - 4,5)](25\pi ) + } \\ {[f( - 3,4,5)](25\pi ) + [f( - 3, - 4,5)](25\pi )} \end{aligned}} \right\}\)

Substitute 7 for \(f(3,4,5),8\) for \(f(3, - 4,5),9\) for \(f( - 3,4,5)\) and 12 for\(f( - 3, - 4,5)\);

\begin{align}& \iint_{H}{f}(x,y,z)dS\cong [(7)(25\pi )+(8)(25\pi )+(9)(25\pi )+(12)(25\pi )] \\&\iint_{H}{f}(x,y,z)dS\cong (25\pi )(7+8+9+12) \\ & \iint_{H}{f}(x,y,z)dS\cong (25\pi )(36) \\&\iint_{H}{f}(x,y,z)dS\cong 2827\\\end{align}

Thus, the value of \(\iint_H f(x,y,z)dS \) is 2827.

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