290 STEP 4. Review the Knowledge You Need to Score High
Example 3
Find the area of the surface generated by revolving about thex-axis the arc defined byx= 3 − 2 t
andy=
√
20 −t^2 when 0≤t≤4.
Step 1. Differentiate
dx
dt
=−2 and
dy
dt
=
−t
√
20 −t^2
.
Step 2. S=
∫ 4
0
2 π
√
20 −t^2
√√
√√
(− 2 )^2 +
(
−t
√
20 −t^2
) 2
dt
= 2 π
∫ 4
0
√
20 −t^2
√
4 +
t^2
20 −t^2
dt
= 2 π
∫ 4
0
√
20 −t^2
√
80 − 3 t^2
20 −t^2
dt
= 2 π
∫ 4
0
√
80 − 3 t^2 dt≈ 2 π( 31. 7768 )≈ 199. 6595
Area and Arc Length for Polar Curves
Area for Polar Curves
Ifr=f(θ) is a continuous polar curve on the intervalα≤θ≤βandα<β<α+ 2 π, then
the area enclosed by the polar curve isA=
1
2
∫β
α
[f(θ)]^2 dθ=
1
2
∫β
α
r^2 dθ.
Example 1
Find the area enclosed byr= 2 +2 cosθon the interval fromθ=0toθ=π.
Step 1. Squarer^2 = 4 +8 cosθ+4 cos^2 θ.
Step 2. A=
1
2
∫π
0
(
4 +8 cosθ+4 cos^2 θ
)
dθ= 2
∫π
0
(
1 +2 cosθ+2 cos^2 θ
)
dθ
= 2
[
2 θ+4 sinθ+ 2
(
θ
2
+
1
4
sin 2θ
)]π
0
= 6 θ+8 sinθ+sin 2θ
∣∣π
0 =^6 π
Arc Length for Polar Curves
For a polar graph defined on a interval (α,β), if the graph does not retrace itself in that
interval and if
dr
dθ
is continuous, then the length of the arc fromθ=αtoθ=βisL=
∫β
α
√
r^2 +
(
dr
dθ
) 2
dθ.