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Chapter 6: Q9P (page 334)

鈭�V脳苍诲蟽over the entire surface of the volume in the first octant bounded byx2+y2+z2=16and the coordinate planes, whereV=x+x2鈭�y2i+(2xyz鈭�2xy)j鈭�xz2k

Short Answer

Expert verified

The solution derived isI=323蟺

Step by step solution

01

Given Information.

The given equation isV=x+x2鈭�y2i+(2xyz鈭�2xy)j鈭�xz2k

02

Definition of vector.

A quantity that has magnitude as well as direction is called a vector. It is typically denoted by an arrow in which the head determines the direction of the vector and the length determines its magnitude.

03

Apply divergence theorem.

Apply the divergence theorem and the fact that鈭�脳V=1+2x+2xz鈭�2x鈭�2xz=1then the integral becomes as shown below.

I=鈭�V鈥�诲蟿

=1843蟺43

=323蟺

Note that only 18of the volume of the sphere is considered because only the first octant is operated.

Hence, the solution derived is I=323蟺.

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