5 Steps to a 5 AP Calculus BC 2019

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.