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Answers without the blur. Sign up and see all textbooks for free! Q. 15

Expert-verified Found in: Page 1119 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # Use what you know about average value from previous sections to propose a formula for the average value of a multivariate function f(x, y, z) on a smooth surface S.

$\mathrm{Hence},\mathrm{the}\mathrm{average}\mathrm{value}\mathrm{of}\mathrm{a}\mathrm{multivariate}\mathrm{function}\mathrm{f}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)\mathrm{on}\mathrm{a}\mathrm{smooth}\mathrm{surface}\mathrm{S}\mathrm{is},\phantom{\rule{0ex}{0ex}}{\mathrm{f}}_{\text{avg}}\left(\mathrm{S}\right)=\frac{{\int }_{\mathrm{S}} \mathrm{f}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)\mathrm{dS}}{{\int }_{\mathrm{S}} 1\mathrm{dS}}$

See the step by step solution

## Step 1. Given

The given function is f(x, y, z) .

## Step 2. Finding the average value of multivariable function

$\mathrm{Surface}\mathrm{integrals}\mathrm{of}\mathrm{multivariate}\mathrm{functions}\mathrm{f}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)\mathrm{is},\phantom{\rule{0ex}{0ex}}\int \mathrm{f}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)\mathrm{dS}.\phantom{\rule{0ex}{0ex}}\mathrm{The}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{smooth}\mathrm{surface}\mathrm{S}\mathrm{is},\phantom{\rule{0ex}{0ex}}{\int }_{\mathrm{S}}1\mathrm{dS}\phantom{\rule{0ex}{0ex}}\mathrm{Hence},\mathrm{the}\mathrm{average}\mathrm{value}\mathrm{of}\mathrm{a}\mathrm{multivariate}\mathrm{function}\mathrm{f}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)\mathrm{on}\mathrm{a}\mathrm{smooth}\mathrm{surface}\mathrm{S}\mathrm{is},\phantom{\rule{0ex}{0ex}}{\mathrm{f}}_{\text{avg}}\left(\mathrm{S}\right)=\frac{{\int }_{\mathrm{S}} \mathrm{f}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)\mathrm{dS}}{{\int }_{\mathrm{S}} 1\mathrm{dS}}$ ### Want to see more solutions like these? 