Free Printable WorksheetMathCalculus IIMath CurriculumAP Calculus BC Unit 6 and Unit 8; FL MA.912.C.4.6, C.4.8-C.4.10; CCSS.MP2, MP3, MP5, MP6; Integrated Calculus II Standards CAL2-IMP, CAL2-NUM; TEKS 19 TAC section 111.55

CAL2.52 Numerical Integration with Simpson's Rule Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.52 Numerical Integration with Simpson's Rule. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.52 Numerical Integration with Simpson's Rule - Select the correct Simpson's Rule approximation and verify that the number of subintervals is appropriate. Standards tags: AP Calculus BC Unit 6 and Unit 8; FL MA.912.C.4.6, C.4.8-C.4.10; CCSS.MP2, MP3, MP5, MP6; Integrated Calculus II Standards CAL2-IMP, CAL2-NUM; TEKS 19 TAC section 111.55.

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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 CAL2.52 Numerical Integration with Simpson's Rule skill assessments:

Problem 1Difficulty: Mastery

A student plans to approximate ∫_0³ (sqrt(1 + sin²(x))/(2 + x)) dx with composite Simpson's Rule using n = 12. Which statement is mathematically correct? Round any approximation to three decimals.

AUse Simpson's Rule: h = 0.25, n = 12 is even, and S_12 ≈ 1.112.

BUse Simpson's Rule with all interior weights equal to 2, giving h/3 times endpoints plus 2 times every interior value.

CDo not use Simpson's Rule because n = 12 is even; use only left rectangles.

DUse h = 0.2727 because Simpson's Rule divides by n - 1.

Solution & Explanation:For composite Simpson's Rule, h = (b - a)/n = 0.25 and n must be even. Since n = 12 is even, the rule applies and the weighted sum gives about 1.112. The divisor is n, not n - 1, and the interior coefficients alternate 4 and 2.

Problem 2Difficulty: Mastery

For ∫_0⁵ R(x) dx using composite Simpson's Rule with n = 10, which setup is correct before evaluation?

A(0.5/3)[R(0) + 4R(0.5) + 4R(1) + 4R(1.5) + 4R(2) + 4R(2.5) + 4R(3) + 4R(3.5) + 4R(4) + 4R(4.5) + R(5)]; this omits the coefficient 2 on even interior points.

B(0.4167/3)[R(0) + 4R(0.5) + 2R(1) + 4R(1.5) + 2R(2) + 4R(2.5) + 2R(3) + 4R(3.5) + 2R(4) + 4R(4.5) + R(5)]; this uses the wrong width.

C(0.5/2)[R(0) + 2R(0.5) + 2R(1) + 2R(1.5) + 2R(2) + 2R(2.5) + 2R(3) + 2R(3.5) + 2R(4) + 2R(4.5) + R(5)]; this is trapezoidal form.

D(0.5/3)[R(0) + 4R(0.5) + 2R(1) + 4R(1.5) + 2R(2) + 4R(2.5) + 2R(3) + 4R(3.5) + 2R(4) + 4R(4.5) + R(5)]; n = 10 is even.

Solution & Explanation:The width is h = (5 - 0)/10 = 0.5, and n = 10 is even. Composite Simpson's Rule uses h/3 times endpoint weights 1, alternating interior weights 4 and 2, and final endpoint weight 1. The trapezoidal form uses h/2 and is a different rule.

Problem 3Difficulty: Mastery

For ∫_-1.5¹.5 H(x) dx using composite Simpson's Rule with n = 12, which setup is correct before evaluation?

A(0.25/3)[H(-1.5) + 4H(-1.25) + 4H(-1) + 4H(-0.75) + 4H(-0.5) + 4H(-0.25) + 4H(0) + 4H(0.25) + 4H(0.5) + 4H(0.75) + 4H(1) + 4H(1.25) + H(1.5)]; this omits the coefficient 2 on even interior points.

B(0.25/3)[H(-1.5) + 4H(-1.25) + 2H(-1) + 4H(-0.75) + 2H(-0.5) + 4H(-0.25) + 2H(0) + 4H(0.25) + 2H(0.5) + 4H(0.75) + 2H(1) + 4H(1.25) + H(1.5)]; n = 12 is even.

C(0.25/2)[H(-1.5) + 2H(-1.25) + 2H(-1) + 2H(-0.75) + 2H(-0.5) + 2H(-0.25) + 2H(0) + 2H(0.25) + 2H(0.5) + 2H(0.75) + 2H(1) + 2H(1.25) + H(1.5)]; this is trapezoidal form.

D(0.2143/3)[H(-1.5) + 4H(-1.25) + 2H(-1) + 4H(-0.75) + 2H(-0.5) + 4H(-0.25) + 2H(0) + 4H(0.25) + 2H(0.5) + 4H(0.75) + 2H(1) + 4H(1.25) + H(1.5)]; this uses the wrong width.

Solution & Explanation:The width is h = (1.5 - -1.5)/12 = 0.25, and n = 12 is even. Composite Simpson's Rule uses h/3 times endpoint weights 1, alternating interior weights 4 and 2, and final endpoint weight 1. The trapezoidal form uses h/2 and is a different rule.

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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 6 and Unit 8; FL MA.912.C.4.6, C.4.8-C.4.10; CCSS.MP2, MP3, MP5, MP6; Integrated Calculus II Standards CAL2-IMP, CAL2-NUM; TEKS 19 TAC section 111.55 (Select the correct Simpson's Rule approximation and verify that the number of subintervals is appropriate. Standards tags: AP Calculus BC Unit 6 and Unit 8; FL MA.912.C.4.6, C.4.8-C.4.10; CCSS.MP2, MP3, MP5, MP6; Integrated Calculus II Standards CAL2-IMP, CAL2-NUM; TEKS 19 TAC section 111.55.).