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Lecture Comments (3)

0 answers

Post by Olivia Brady on May 11 at 11:11:50 PM

Im slightly confused why there were two 1/2's multiplied in the last example

1 answer

Last reply by: Jingwei Xie
Sun Jan 18, 2015 3:36 PM

Post by Thomas Zhang on March 17, 2014

Hey John, you forgot the -1 in the first example, just in case anyone was wondering. So it's -1/4

Area for Parametric & Polar Curves

  • For parametric function,
  • For polar function,
  • Apply directly with correct bounds

Area for Parametric & Polar Curves

Find the area bounded by the parametric curve defined by \protect
x(t) = t − 7
y(t) = t + 8
\protect
from − 2 to 2
  • Find [dx/dt]
  • x′ = dt
A = ∫− 22 t + 8dt = [ [(t2)/2] + 8t ]−2 2 = 32
Find the area bounded by the parametric curve defined by \protect
x(t) = 2t
y(t) = t3 − 6t\protect
,from 0 to 10
  • Find [dx/dt]
  • x′ = 2dt
  • Apply Area Formula
  • A = ∫010 ( t3 − 6t )( 2dt )
  • = 2∫010 ( t3 − 6t )( dt )
  • = 2( ∫010 t3dt − 6∫010 dt )
  • = 2( [ [(t4)/4] ]010 − 6[ t ]010 )
  • = 2( 1400 )
= 2800
Find the area bounded by the parametric curve defined by \protect
x(t) = t2
y(t) = ( t − 1 )2\protect
,from − 7 to 0
  • Find [dx/dt]
  • x′ = 2t dt
  • Apply the Area Formula
  • A = ∫ − 70 ( t − 1 )2(2t dt)
  • = 2∫ − 70 t( t − 1 )2dt
= 2∫ − 70 t3 − 2t2 + tdt = 2[ [(t4)/4] − [2/3]t3 + [(t2)/2] ] − 70 = − 1706.8
Find the area bounded by the parametric curve defined by \protect
x(t) = (t − 1)2
y(t) = t2\protect
,from 0 to 7
  • Find [dx/dt]
  • x′ = 2t − 2 dt
  • Apply the Area Formula
  • A = ∫07 ( t2 )( 2t − 2 dt )
  • = 2∫07 ( t2 )(t − 1)dt
  • = 2∫07 t3 − t2dt
  • = 2( [ [(t4)/4] ]07 − [ [(t3)/3] ]07 )
= 485.9
Find the area bounded by the parametric curve defined by \protect
x(t) = 3t
y(t) = − sint\protect
,from 0 to π
  • Find [dx/dt]
  • x′ = 3 dt
  • Apply the Area Formula
= − 3∫0π tsint dt = − 3[ sint − tcost ]0π = − 3π
Find the area bounded by the polar curve defined by r = cosθ,from − π to [(7π)/4]
  • Find [dr/(dθ)]
  • r′ = − sinθdθ
  • Apply the Area Formula
  • A = [1/2]∫ − π[(7π)/4] ( cosθ )2( − sinθ ) dθ
  • Use chain rule to integrate
  • [1/2]∫ − π[(7π)/4] ( cosθ )2( − sinθ ) dθ = [1/2][ [(cos3θ)/3] ] − π[(7π)/4]
  • = [(4 + √2 )/24]
= 0.23
Find the area bounded by the polar curve defined by r = sin2θ , from 0 to [(2π)/3]
  • Find [dr/(dθ)]
  • r′ = 2cos2θdθ
  • Apply the Area Formula
  • A = [1/2]∫0[(2π)/3] ( sin2θ )2(2cos2θdθ)
  • = ∫0[(2π)/3] sin22θcos2θdθ
  • Use chain rule to integrate
= [ [1/6]sin3(2θ) ]0[(2π)/3] = − [(√3 )/16] = − .11
Find the area bounded by the polar curve defined by r = 2e − θ, from 0 to 5
  • Find [dr/(dθ)]
  • r′ = − 2e − θ
  • Apply Area Formula
  • A = [1/2]∫05 ( 2e − θ )2( − 2e − θ ) dθ
  • = − 2∫05 ( e − 2θ )( e − θ ) dθ
  • = − 2∫05 e − 3θ
  • = [ [2/3]e − 3θ ]05
  • = [2/3]( e − 15 − 1 )
= − 0.67
Find the area bounded by the polar curve defined by r = secθ , from − [(5π)/6] to 0
  • Find [dr/(dθ)]
  • r′ = tanθsecθdθ
  • Apply the Area Formula
  • A = [1/2]∫ − [(5π)/6]0 ( secθ ) 2tanθsecθdθ
  • = [1/2]∫ − [(5π)/6]0 [(sinθ)/(cos4θ)] dθ
Integrate using Chain Rule
[1/2]∫ − [(5π)/6]0 [(sinθ)/(cos4θ)] dθ = [1/2][ [(sec3θ)/3] ] − [(5π)/6]0 = − 0.42
Find the area bounded by the polar curve defined by r = tanθ , from [( − π)/3] to 0
  • Find [dr/(dθ)]
  • r′ = sec2θdθ
  • Apply the Area Formula
  • A = [1/2]∫[( − π)/3]0 ( tanθ )2( sec2θ ) dθ
  • Use Pythagorean identity to alter function
  • [1/2]∫[( − π)/3]0 ( tanθ )2( sec2θ ) dθ = [1/2]∫[( − π)/3]0 ( sec2θ− 1 ) sec2θdθ
  • = [1/2]∫[( − π)/3]0 sec4θ− sec2θ dθ
  • = [1/2]∫[( − π)/3]0 sec4θdθ− ∫[( − π)/3]0 sec2θ dθ
  • = [1/2]( [ [(sec2θtanθ)/3] ][( − π)/3]0 − [ tanθ ][( − π)/3]0 )
  • = [(√3 )/2]
= 0.87

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Area for Parametric & Polar Curves

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Area for Parametric Curves: Parametric Function 0:10
  • Example 1: Area for Parametric Curves 0:35
  • Area for Parametric Curves: Polar Function 2:50
  • Example 2: Area for Polar Curves 3:18