Connecting...

This is a quick preview of the lesson. For full access, please Log In or Sign up.
For more information, please see full course syllabus of Calculus BC

Calculus BC Area for Parametric & Polar Curves

Section 4: Applications of Integration: Lecture 3 | 4:33 min

Lecture Description

Next Lecture

Previous Lecture

Discussion
Answer Engine
Study Guides
Practice Questions
Download Lecture Slides
Table of Contents
Related Books

Lecture Comments (3)

0 answers

Post by Olivia Brady on May 11, 2016

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

Answer EngineGet answers to any question!Ask any question related to Calculus BC

Working on the solution...

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 = ∫_{− 2}² 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 = ∫_{− 7}⁰ ( t − 1 )²(2t dt)
= 2∫_{− 7}⁰ t( t − 1 )²dt

= 2∫_{− 7}⁰ t³ − 2t² + tdt = 2[ [(t⁴)/4] − [2/3]t³ + [(t²)/2] ]_{− 7}⁰ = − 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∫₀^π tsint dt = − 3[ sint − tcost ]₀^π = − 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} dθ
Apply Area Formula
A = [1/2]∫₀⁵ ( 2e^{− θ} )²( − 2e^{− θ} ) dθ
= − 2∫₀⁵ ( e^{− 2θ} )( e^{− θ} ) dθ
= − 2∫₀⁵ e^{− 3θ} dθ
= [ [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

« Previous Question | Next Question »

*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

Area for Parametric Curves: Parametric Function

Example 1: Area for Parametric Curves

Area for Parametric Curves: Polar Function

Example 2: Area for Polar Curves

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

Mathematics: AP Calculus BC

Section 1: Prerequisites
	Parametric Curves	10:10
	Polar Coordinates	14:54
	Vector Functions	12:05
Section 2: Differentiation
	Parametric Differentiation	13:15
	Polar Differentiation	10:14
	L'Hopital's Rule	6:43
Section 3: Integration
	Integration By Parts	19:04
	Integration By Partial Fractions	22:04
	Improper Integrals	19:01
Section 4: Applications of Integration
	Logistic Growth	19:16
	Arc Length for Parametric & Polar Curves	17:40
	Area for Parametric & Polar Curves	4:33
Section 5: Sequences, Series, & Approximations
	Definition & Convergence	7:10
	Geometric Series	10:59
	Harmonic & P Series	3:58
	Integral Test	11:31
	Comparison Test	7:47
	Limit Comparison Test	9:40
	Ratio Test	11:20
	Alternating Series	9:11
	Absolute Convergence	7:13
	Power Series Convergence	12:52
	Taylor Series	15:11
	Power Series Operations	16:40
	Lagrange Error	7:09
Section 6: Practice Test
	AP Calc BC Practice Test Section 1: Multi Choice Part 1	15:45
	AP Calc BC Practice Test Section 1: Multi Choice Part 2	21:38
	AP Calc BC Practice Test Section 1: Multi Choice Part 3	17:31
	AP Calc BC Practice Test Section 1: Multi Choice Part 4	14:17
	AP Calc BC Practice Test Section 1: Multi Choice Part 5	22:45
	AP Calc BC Practice Test Section 1: Free Response Part 1	9:55
	AP Calc BC Practice Test Section 1: Free Response Part 2	10:08
	AP Calc BC Practice Test Section 1: Free Response Part 3	13:58

Related Books

	Cracking the AP Calculus BC Exam, 2015 Edition Author: Princeton Review ISBN: 804124825 Publisher: Princeton Review Year: 2014 The book includes 3 full length practice tests with detailed explanations, a review of all the key concepts, and targeted strateeies to ace th exam for your highest score.
	Cliffs AP: Calculus AB & BC, 3rd Edition Authors: Dale W Johnson, Kerry J King ISBN: 764586831 Publisher: Cliffs Notes Year: 2001 The book has a topic-by topic breakdown and lots of problem approach suggestions for both free response and multiple choice calculus questions. The book also includes two full length tests.

Related Books

Name	Description	Link
BookRenter.com	BookRenter.com is simply the most reliable online textbook rental service.	Visit BookRenter.com
PhysicsForums.com Homework Help	Physics Forums is a scientific community for students looking for math & science help.	Visit PhysicsForums.com Homework Help

John Zhu

Area for Parametric & Polar Curves

Slide Duration:

Table of Contents

Section 1: Prerequisites

Parametric Curves

10m 10s

Intro