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

CAL2.64 Volumes with Cross Sections Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.64 Volumes with Cross Sections. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.64 Volumes with Cross-Sections - Select the correct cross-sectional area function and volume integral for squares, rectangles, triangles, or semicircles. 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 CAL2.64 Volumes with Cross Sections skill assessments:

Problem 1Difficulty: Mastery

The base of a solid is the region bounded by y=x and y=x^(1/3) for 0≤x≤1. Cross-sections perpendicular to the x-axis are squares whose side is the segment in the base region. Which cross-sectional area function and volume integral are correct?

AA(x)=(π/8)(x^(1/3)-x)²; V=∫_{0}¹ (π/8)(x^(1/3)-x)² dx

BA(x)=(x^(1/3)+x)²; V=∫_{0}¹ (x^(1/3)+x)² dx

CA(y)=(y^(1/3)-y)²; V=∫_{0}¹ (y^(1/3)-y)² dy

DA(x)=(x^(1/3)-x)²; V=∫_{0}¹ (x^(1/3)-x)² dx

Solution & Explanation:A slice perpendicular to the x-axis uses a vertical segment, so the slice length is s(x)=x^(1/3)-x by top-minus-bottom. A square with side length s has area s². Therefore A(x)=(x^(1/3)-x)² and V=∫_{0}¹ (x^(1/3)-x)² dx. A common mistake is to use x^(1/3)+x or the wrong differential, which does not match the stated slices.

Problem 2Difficulty: Mastery

The base of a solid is the region bounded by x=0 and x=ln(1+y) for 0≤y≤e-1. Cross-sections perpendicular to the y-axis are triangles whose altitude is 1/2 times the base segment in the region. Which cross-sectional area function and volume integral are correct?

AA(y)=(1/2)(ln(1+y))²; V=∫_{0}^e-1 (1/2)(ln(1+y))² dy

BA(y)=(1/4)(y)²; V=∫_{0}^e-1 (1/4)(y)² dy

CA(y)=(1/4)(ln(1+y))²; V=∫_{0}^e-1 (1/4)(ln(1+y))² dy

DA(x)=(1/4)(ln(1+x))²; V=∫_{0}^e-1 (1/4)(ln(1+x))² dx

Solution & Explanation:A slice perpendicular to the y-axis uses a horizontal segment, so the slice length is s(y)=ln(1+y) by right-minus-left. Triangle area is (1/2)(base)(altitude); with altitude 1/2s, A = 1/4s². Therefore A(y)=(1/4)(ln(1+y))² and V=∫_{0}^e-1 (1/4)(ln(1+y))² dy. A common mistake is to use y or the wrong differential, which does not match the stated slices.

Problem 3Difficulty: Mastery

The base of a solid is the region bounded by y=1 and y=4/(x+2) for 0≤x≤2. Cross-sections perpendicular to the x-axis are equilateral triangles whose side is the segment in the base region. Which cross-sectional area function and volume integral are correct?

AA(x)=(1/2)(4/(x+2)-1)²; V=∫_{0}² (1/2)(4/(x+2)-1)² dx

BA(x)=(sqrt(3)/4)(4/(x+2)-1)²; V=∫_{0}² (sqrt(3)/4)(4/(x+2)-1)² dx

CA(x)=(sqrt(3)/4)(4/(x+2)+1)²; V=∫_{0}² (sqrt(3)/4)(4/(x+2)+1)² dx

DA(y)=(sqrt(3)/4)(4/(y+2)-1)²; V=∫_{0}² (sqrt(3)/4)(4/(y+2)-1)² dy

Solution & Explanation:A slice perpendicular to the x-axis uses a vertical segment, so the slice length is s(x)=4/(x+2)-1 by top-minus-bottom. An equilateral triangle with side length s has area (sqrt(3)/4)s². Therefore A(x)=(sqrt(3)/4)(4/(x+2)-1)² and V=∫_{0}² (sqrt(3)/4)(4/(x+2)-1)² dx. A common mistake is to use 4/(x+2)+1 or the wrong differential, which does not match the stated slices.

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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 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 cross-sectional area function and volume integral for squares, rectangles, triangles, or semicircles. 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).