Free Printable WorksheetMathCalculus IIMath CurriculumAP Calculus BC Unit 9; CCSS.HSF.IF, HSG.GPE, HSN.VM extension; CCSS.MP2, MP4, MP5, MP6, MP7; Integrated Calculus II Standards CAL2-PPV; TEKS 19 TAC section 111.55

CAL2.109 Area Between Polar Curves Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.109 Area Between Polar Curves. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.109 Area Between Polar Curves - Choose the correct bounds and integrand for area between two polar curves. Standards: AP Calculus BC Unit 9; CCSS.HSF.IF, HSG.GPE, HSN.VM extension; CCSS.MP2, MP4, MP5, MP6, MP7; Integrated Calculus II Standards CAL2-PPV; TEKS 19 TAC section 111.55

Whether you are an educator preparing classroom handouts, a tutor reinforcing weekly topics, or a parent supporting homeschool learning, this printable resource gives students the deliberate practice required to achieve subject fluency and master state standards.

Grade

Calculus II

Subject

Math

Domain

Math Curriculum

Formats

PDF & DOCX

Sample Practice Problems

Preview the types of problems and questions covered in this worksheet

Below are sample questions representative of the exercises included on this worksheet and related CAL2.109 Area Between Polar Curves skill assessments:

Problem 1Difficulty: Mastery

On 0 ≤ θ ≤ π/2, the curves r = sqrt(4 + θ) and r = sqrt(6 - θ) intersect at θ = 1. Which setup represents the total area between them on this interval?

A[∫_{0}¹ [(sqrt(6 - θ))² - (sqrt(4 + θ))²] dθ + ∫_{1}^π/2 [(sqrt(4 + θ))² - (sqrt(6 - θ))²] dθ]

B(1/2)[∫_{0}¹ [(sqrt(6 - θ))² - (sqrt(4 + θ))²] dθ + ∫_{1}^π/2 [(sqrt(4 + θ))² - (sqrt(6 - θ))²] dθ]

C(1/2)[∫_{0}¹ [(sqrt(4 + θ))² - (sqrt(6 - θ))²] dθ + ∫_{1}^π/2 [(sqrt(6 - θ))² - (sqrt(4 + θ))²] dθ]

D(1/2)∫_{0}^π/2 [(sqrt(6 - θ))² - (sqrt(4 + θ))²] dθ

Solution & Explanation:A polar area setup uses (1/2)∫(outer² - inner²)dθ. Compare squared radii: 6 - θ is larger before θ = 1, and 4 + θ is larger after θ = 1. Because the outer curve changes, the integral must be split; a single signed difference can cancel part of the area.

Problem 2Difficulty: Mastery

On -π/2 ≤ θ ≤ π/2, compare r = 3 + cos θ and r = 3 - cos θ. Which setup represents the area between them?

A∫_{-π/2}^π/2 [(3 + cos θ)² - (3 - cos θ)²] dθ

B(1/2)∫_{-π/2}^π/2 [(3 - cos θ)² - (3 + cos θ)²] dθ

C(1/2)∫_{-π/2}^π/2 [(3 + cos θ)² - (3 - cos θ)²] dθ

D(1/2)∫_{0}^π/2 [(3 + cos θ)² - (3 - cos θ)²] dθ

Solution & Explanation:Use A = (1/2)∫(r_outer² - r_inner²)dθ. Since cos θ ≥ 0 on the entire interval, r = 3 + cos θ stays outside and no split is needed despite the similar forms. The factor 1/2 and the outer-minus-inner order are both required.

Problem 3Difficulty: Mastery

On -π/2 ≤ θ ≤ π/2, compare r = 3 + sin θ and r = 3 - sin θ. Which setup gives the total area between them?

A[∫_{-π/2}⁰ [(3 - sin θ)² - (3 + sin θ)²] dθ + ∫_{0}^π/2 [(3 + sin θ)² - (3 - sin θ)²] dθ]

B(1/2)[∫_{-π/2}⁰ [(3 + sin θ)² - (3 - sin θ)²] dθ + ∫_{0}^π/2 [(3 - sin θ)² - (3 + sin θ)²] dθ]

C(1/2)∫_{-π/2}^π/2 [(3 + sin θ)² - (3 - sin θ)²] dθ

D(1/2)[∫_{-π/2}⁰ [(3 - sin θ)² - (3 + sin θ)²] dθ + ∫_{0}^π/2 [(3 + sin θ)² - (3 - sin θ)²] dθ]

Solution & Explanation:A polar area setup uses (1/2)∫(outer² - inner²)dθ. The sign of sin θ changes at 0, so the outer curve switches at the polar axis. Because the outer curve changes, the integral must be split; a single signed difference can cancel part of the area.

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QWhat grade level and standards does this worksheet align with?

This worksheet is designed specifically for Calculus II Math students. It aligns with educational standard AP Calculus BC Unit 9; CCSS.HSF.IF, HSG.GPE, HSN.VM extension; CCSS.MP2, MP4, MP5, MP6, MP7; Integrated Calculus II Standards CAL2-PPV; TEKS 19 TAC section 111.55 (Choose the correct bounds and integrand for area between two polar curves. Standards: AP Calculus BC Unit 9; CCSS.HSF.IF, HSG.GPE, HSN.VM extension; CCSS.MP2, MP4, MP5, MP6, MP7; Integrated Calculus II Standards CAL2-PPV; TEKS 19 TAC section 111.55).