MATH 234 HW 5 Answer
Section 15.6
[16]. Find the iterated integral for evaluating R R RDf(r, θ, z)dz rdrdθ over the region D.
Dis the right circular cylinder whose base is the circle r= 3 cos θand whose top lies in the plane z= 5−x.
Solution:
Zπ
0Z3 cos θ
0Z5−rcos θ
0
f(r, θ, z)dzr drdθ =Zπ
2
−π
2Z3 cos θ
0Z5−rcos θ
0
f(r, θ, z)dzr drdθ
[34]. The solid bounded below by the hemisphere ρ= 1, z≥0,
and bounded above by the cardioid of revolution ρ= 1 + cos φ.
(a) Find the spherical coordinate limits for the integral that calculates the volume of the solid.
(b) Evalute the integral obtained from (a).
Solution:
Z2π
0Zπ
2
0Z1+cos φ
1
ρ2sin φdρdφdθ
=Z2π
0Zπ
2
01
3ρ31+cos φ
1
sin φdφdθ =1
3Z2π
0Zπ
2
03 cos φ+ 3 cos2φ+ cos3φsin φdφdθ
=1
3Z2π
0
−3
2cos2φ+ cos3φ+1
4cos4φ
π
2
0
dθ =1
3Z2π
0
11
4dθ =11
6π
Section 16.1
[11]. Evaluate RC(xy +y+z)ds along the curve −→
r(t) = h2t, t, 2−2tifor 0 ≤t≤1.
Solution: (xy +y+z) = (2t) (t)+(t) + (2 −2t) = 2t2−t+ 2
−→
r0(t) = h2,1,−2i=⇒ |−→
r0(t)|= 3
RC(xy +y+z)ds =R1
02t2−t+ 2·3dt =R1
06t2−3t+ 6dt =2t3−3
2t2+ 6t1
0=13
2
[21]. Integrate f(x, y) = x+yover the curve C:x2+y2= 4 in the first quadrant from (2,0) to (0,2).
Solution: Parametrize Cby −→
r(t) = h2 cos t, 2 sin tifor 0 ≤t≤π
2.
=⇒−→
r0(t) = h−2 sin t, 2 cos ti=⇒ |−→
r0(t)|= 2
=⇒f(t) = 2 cos t+ 2 sin t
RCf(x, y)ds =Rπ
2
0(2 cos t+ 2 sin t)·2dt = 4 Rπ
2
0(cos t+ sin t)dt = 4 [sin t−cos t]
π
2
0= 8
1