Free Printable WorksheetMathCalculus IIMath CurriculumAP Calculus BC Unit 8; FL MA.912.C.5.5-C.5.7; CCSS.HSM; CCSS.HSN.Q.1-3; CCSS.MP2, MP4, MP6, MP7; Integrated Calculus II Standards CAL2-APP; TEKS 19 TAC section 111.55

CAL2.68 Volumes Around Non Axis Lines Worksheet

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Q: What 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 8; FL MA.912.C.5.5-C.5.7; CCSS.HSM; CCSS.HSN.Q.1-3; CCSS.MP2, MP4, MP6, MP7; Integrated Calculus II Standards CAL2-APP; TEKS 19 TAC section 111.55 (Select the correct radius expression and method for rotation around a horizontal or vertical line other than an axis. Standards: AP Calculus BC Unit 8; FL MA.912.C.5.5-C.5.7; CCSS.HSM; CCSS.HSN.Q.1-3; CCSS.MP2, MP4, MP6, MP7; Integrated Calculus II Standards CAL2-APP; TEKS 19 TAC section 111.55).

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.68 Volumes Around Non Axis Lines. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.68 Volumes Around Non-Axis Lines - Select the correct radius expression and method for rotation around a horizontal or vertical line other than an axis. Standards: AP Calculus BC Unit 8; FL MA.912.C.5.5-C.5.7; CCSS.HSM; CCSS.HSN.Q.1-3; CCSS.MP2, MP4, MP6, MP7; Integrated Calculus II Standards CAL2-APP; 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 CAL2.68 Volumes Around Non Axis Lines skill assessments:

Problem 1Difficulty: Mastery

Region R is bounded by y = 0 and y = 4 - x² for -2 <= x <= 2. R is rotated about y = 2. Which washer/disk setup is correct?

AV = π∫_-2² [2² - (2 - (4 - x²))²] dx.

BV = π∫_-2² (2 - x²)² dx.

CV = 2π∫_-2² ∣x - 2∣(4 - x²) dx.

DV = π[∫_-2^(-sqrt(2)) (2² - (x² - 2)²) dx + ∫_(-sqrt(2))^(sqrt(2)) 2² dx + ∫_(sqrt(2))² (2² - (x² - 2)²) dx].

Solution & Explanation:For ∣x∣ <= sqrt(2), the slice contains y = 2 and the bottom edge y = 0 is at least as far as the top edge, so the cross-section is a disk of radius 2. For ∣x∣ > sqrt(2), the slice lies below y = 2 and becomes a washer with inner radius x² - 2.

Problem 2Difficulty: Mastery

For 0 <= y <= 2, region R lies between x = y and x = y + 3. R is rotated about x = 2. Which disk setup is correct?

AV = π∫_0² [(2 - y)² - (2 - (y + 3))²] dy.

BV = π∫_0² (y + 1)² dy.

CV = π[∫_0^(1/2) (2 - y)² dy + ∫_(1/2)² (y + 1)² dy].

DV = 2π∫_0² ∣y - 2∣((y + 3) - y) dy.

Solution & Explanation:The vertical axis x = 2 lies inside each horizontal slice. The left distance is 2 - y and the right distance is y + 1; they are equal at y = 1/2, so the disk radius changes there.

Problem 3Difficulty: Mastery

Region R is bounded by y = x and y = x + 3 for 0 <= x <= 2. R is rotated about y = 2. Which disk setup is correct?

AV = π∫_0² [(2 - x)² - (2 - (x + 3))²] dx.

BV = π[∫_0^(1/2) (2 - x)² dx + ∫_(1/2)² (x + 1)² dx].

CV = π∫_0² (x + 1)² dx.

DV = 2π∫_0² ∣x - 2∣((x + 3) - x) dx.

Solution & Explanation:The axis y = 2 lies inside the vertical slice. The bottom distance is 2 - x and the top distance is x + 1. These are equal at x = 1/2, so the disk radius switches there.

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Frequently Asked Questions

Common questions about printing, using, and solving this worksheet

QHow do I download and print this CAL2.68 Volumes Around Non Axis Lines worksheet?

Click either the "Download PDF" or "Download DOCX" button at the top of the page. You can print the PDF directly from your browser or edit the DOCX file in Microsoft Word or Google Docs to tailor the questions to your students.

QIs this CAL2.68 Volumes Around Non Axis Lines worksheet free for teachers and homeschoolers?

Yes! All learning materials, worksheets, and practice resources on Free K12 Practice are 100% free for teachers, students, parents, tutors, and homeschooling families for non-commercial educational use. No account or subscription is required.

QCan students practice CAL2.68 Volumes Around Non Axis Lines online with instant feedback?

Yes! In addition to this printable worksheet, students can practice CAL2.68 Volumes Around Non Axis Lines interactively on our platform with automated grading, instant score tracking, and detailed step-by-step solutions.

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 8; FL MA.912.C.5.5-C.5.7; CCSS.HSM; CCSS.HSN.Q.1-3; CCSS.MP2, MP4, MP6, MP7; Integrated Calculus II Standards CAL2-APP; TEKS 19 TAC section 111.55 (Select the correct radius expression and method for rotation around a horizontal or vertical line other than an axis. Standards: AP Calculus BC Unit 8; FL MA.912.C.5.5-C.5.7; CCSS.HSM; CCSS.HSN.Q.1-3; CCSS.MP2, MP4, MP6, MP7; Integrated Calculus II Standards CAL2-APP; TEKS 19 TAC section 111.55).