Areas, Volumes, and Arc Lengths 289
12.5 Integration of Parametric, Polar, and Vector Curves
Main Concepts:Area, Arc Length, and Surface Area for Parametric Curves; Area and Arc
Length for Polar Curves; Integration of a Vector-Valued Function
Area, Arc Length, and Surface Area for Parametric Curves
Area for Parametric Curves
For a curve defined parametrically byx=f(t) andy=g(t), the area bounded by the portion
of the curve betweent=αandt=βisA=
∫β
α
g(t)f′(t)dt.
Example 1
Find the area bounded byx=2 sint,y=3 sin^2 t.
Step 1. Determine the limits of integration. The symmetry of the graph allows us to
integrate fromt=0tot=π/2 and multiply by 2.
Step 2. Differentiate
dx
dt
=2 cost.
Step 3. A= 2
∫π/ 2
0
3 sin^2 t(2 cost)dt= 12
∫π/ 2
0
(sin^2 tcost)dt=4 sin^3 t
∣
∣∣π/^2
0
= 4
Arc Length for Parametric Curves
The length of that arc isL=
∫β
α
√(
dx
dt
) 2
+
(
dy
dt
) 2
dt.
Example 2
Find the length of the arc defined byx=etcostandy=etsintfromt=0tot=4.
Step 1. Differentiate
dx
dt
=etcost−etsintand
dy
dt
=etcost+etsint.
Step 2. L=
∫ 4
0
√
(etcost−etsint)^2 +(etcost+etsint)^2 dt
L=
∫ 4
0
√
2 e^2 t(cos^2 t+sin^2 t)dt=
∫ 4
0
√
2 e^2 tdt=
√
2
∫ 4
0
etdt=
√
2 et
∣∣
∣
4
0
=
√
2 e^4 −
√
2
Surface Area for Parametric Curves
The surface area created when that arc is revolved about thex-axis is
S=
∫β
α
2 πy
√(
dx
dt
) 2
+
(
dy
dt
) 2
dt.