Answered: The solid E is bounded by the surfaces…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

The solid E is bounded by the surfaces z = 0 and z = 25 - x^2 - y^2. Set up triple integrals to find the volume of the solid E.

Expert Solution

Step 1: Set up triple integrals for the volume of the solid E.

From the given data the solid E is bounded by the surfaces z = 0 and $z space equals space 25 space minus space x squared minus space y squared$ .

so the z limit is given by $z space equals space 0 space rightwards arrow space space 25 space minus space x squared minus space y squared$

If z = 0 then $x squared plus space y squared space equals space 25$

this is a circle at center origin and radius 5.

The polar coordinate form of the circle is

$x space equals space r space cos open parentheses theta close parentheses$ , $y space equals space r space sin open parentheses theta close parentheses$ and $d x space d y space equals space r space d r space d theta$ then $x squared space plus space y squared space equals space r squared$

where $r space equals space 0 space rightwards arrow space 5$ and

$theta space equals space 0 space t o space 2 straight pi$

the volume of the solid E in triple integrals form as follows:

$V space equals space integral subscript x space integral subscript y space integral subscript z equals 0 end subscript superscript 25 space minus space x squared minus space y squared end superscript space d z space d y space d x$