Evaluate the integral below, where B is the ball with center the origin and radius 2. II, (x² + y? + z?)² dV
Evaluate the integral below, where B is the ball with center the origin and radius 2. II, (x² + y? + z?)² dV
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Evaluate the integral below, where B is the ball with center the origin and radius 2.
III x? + y² + z²)2 dv
Step
1
In spherical coordinates, (x2 + y2 + z?)2 = p
and
dV -
dp de do.
p² sin (o)
Step 2
TI| x? + y? +
- z?)² dV =
Now we have
sin(4) dp d0 do. The region B is the ball centered at
JB
the origin with radius 2. Thus, we have
sin(4) dp de dø =
p° sin(4) dp de dộ.
Step 3
The triple integral can be rewritten as a product of single integrals, as follows.
po sin(4) dp de do
[ sin ()
do ·
de ·
dp
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