Multivariate Calculus

Running head: MULTIVARIATE CALCULUS 1
Multivariate Calculus
Name
Institution

MULTIVARIATE CALCULUS 2
Multivariate Calculus
Question 1
Parabolic cylinder z=4−x2
Planes x=0 , y=6 , z=0
The density of the body is 1. Therefore, mass=volume
z=4−x2 =0 , 4=x2 , x=2
0 ≤ x ≤ 2 ,0 ≤ y ≤6 ,∧0 ≤ z ≤ 4−x2
Mass=volume=∫
0
2
∫
0
6
∫
0
4 −x2
dzdydx
∫
0
4 −x2
dz= [ z ]0
4− x2
=4−z2
∫
0
6
4−x2 dy = [ (4−x2 ) y ] 0
6
=6(4−x2)
∫
0
2
6 (4−x2 )dx =6 [(4 x− x
3
3
) ]0
2
6 ( 4 ( 2 ) − 2
3
3
)=32
Question 2
Since the double cone is twice a single cone, we work with the region R(Z >0) then double the
result. The region formed is shown in the figure below.

MULTIVARIATE CALCULUS 3
We know, r2=x2 + y2 + z2
¿ 1 ≤r 2 ≤ 4 we get 1≤ r ≤2.
0 ≤ θ ≤2 π ,0 ≤ ∅ ≤ π
4 ,∧z =rcos ∅
∫∫∫
V
❑
z2 dv=∫
1
2
∫
0
2 π
∫
0
π
4
r4 cos2 ∅ sin ∅ d ∅ dθdr
Let cos ∅=u , d ∅= −1
sin ∅ du
∫
0
π
4
r 4 cos2 ∅sin ∅ d ∅=¿ r4
∫
0
π
4
u2 sin ∅( −1
sin∅ du)¿
¿−r 4
∫
0
π
4
u2 du=−r4
| u3
3 |0
π
4