Free Printable WorksheetMathCalculus IIMath CurriculumAP Calculus BC Unit 10; CCSS.HSA.SSE, HSF.BF; CCSS.MP2, MP5, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-PS; TEKS 19 TAC section 111.55

CAL2.151 Approximating Definite Integrals with Series Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.151 Approximating Definite Integrals with Series. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.151 Approximating Definite Integrals with Series - Select the series method and error estimate for approximating a non-elementary definite integral. Standards: AP Calculus BC Unit 10; CCSS.HSA.SSE, HSF.BF; CCSS.MP2, MP5, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-PS; 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

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Below are sample questions representative of the exercises included on this worksheet and related CAL2.151 Approximating Definite Integrals with Series skill assessments:

Problem 1Difficulty: Mastery

Using the alternating-series estimate, what is the smallest number of nonzero integrated terms needed to approximate I=∫_0¹ (1-e^-x) / x dx, with the integrand defined by its removable value at x=0 with error less than 1e-7?

A7 nonzero terms

B10 nonzero terms

C8 nonzero terms

D9 nonzero terms

Solution & Explanation:After 9 terms, the first omitted integrated term is 1¹⁰/(10·10!)≈2.7557e-08, which is below 1e-7. With fewer terms, the first omitted term would be about 3.0619e-07 if only 8 terms were used, so 9 is the smallest valid count.

Problem 2Difficulty: Mastery

Using the alternating-series estimate, what is the smallest number of nonzero integrated terms needed to approximate I=∫_0¹ (1-cos x)/x² dx, with the integrand defined by its removable value at x=0 with error less than 1e-7?

A2 nonzero terms

B3 nonzero terms

C4 nonzero terms

D5 nonzero terms

Solution & Explanation:After 4 terms, the first omitted integrated term is 1⁹/(10!·9)≈3.0619e-08, which is below 1e-7. With fewer terms, the first omitted term would be about 3.5431e-06 if only 3 terms were used, so 4 is the smallest valid count.

Problem 3Difficulty: Mastery

Using the alternating-series estimate, what is the smallest number of nonzero integrated terms needed to approximate I=∫_0¹ x cos(x³) dx with error less than 1e-6?

A4 nonzero terms

B3 nonzero terms

C5 nonzero terms

D2 nonzero terms

Solution & Explanation:After 4 terms, the first omitted integrated term is 1²⁶/(8!·26)≈9.5391e-07, which is below 1e-6. With fewer terms, the first omitted term would be about 6.94444e-05 if only 3 terms were used, so 4 is the smallest valid count.

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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 10; CCSS.HSA.SSE, HSF.BF; CCSS.MP2, MP5, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-PS; TEKS 19 TAC section 111.55 (Select the series method and error estimate for approximating a non-elementary definite integral. Standards: AP Calculus BC Unit 10; CCSS.HSA.SSE, HSF.BF; CCSS.MP2, MP5, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-PS; TEKS 19 TAC section 111.55).