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

CAL2.76 Modeling with Integral Applications Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.76 Modeling with Integral Applications. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.76 Modeling with Integral Applications - Choose the integral model that best fits a described physical, biological, or economic situation. 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

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Below are sample questions representative of the exercises included on this worksheet and related CAL2.76 Modeling with Integral Applications skill assessments:

Problem 1Difficulty: Mastery
A drone follows a parametric path x=x(t), y=y(t), z=z(t) for a <= t <= b. Its fuel use is c(t) liters per meter traveled. Which integral gives the total fuel used?
A∫_ab c(t) dt
B∫_ab sqrt((x')2+(y')2+(z')2) dt
C∫_ab c(t)sqrt((x'(t))2+(y'(t))2+(z'(t))2) dt
D∫_ab c(t)[x'(t)+y'(t)+z'(t)] dt
Solution & Explanation:Fuel use per meter must be multiplied by the arclength element ds=sqrt((x')2+(y')2+(z')2)dt. Integrating c(t) over time would require units of liters per second, not liters per meter.
Problem 2Difficulty: Mastery
Capital K(t) depreciates continuously at rate delta and receives investment flow I(s). Which model gives capital at time T from initial capital K0 at time 0 and investments over 0 <= s <= T?
AK0+∫_0T I(s)e-delta s ds
BK0e-delta T+∫_0T I(s)e-delta(T-s) ds
CK0edelta T+∫_0T I(s)edelta(T-s) ds
D∫_0T [K0+I(s)]e-delta s ds
Solution & Explanation:Initial capital depreciates for T years, giving K0e-delta T. An investment made at time s depreciates only from s to T, so its remaining amount is weighted by e-delta(T-s).
Problem 3Difficulty: Mastery
At time t, pollutant concentration in a pipe is C(t) milligrams per liter and flow rate is Q(t) liters per second. Which integral gives the total pollutant mass passing the sensor from t=0 to t=T?
A∫_0T C(t)Q(t) dt
B∫_0T [C(t)+Q(t)] dt
C(1/T)∫_0T C(t)Q(t) dt
D∫_0T C'(t)Q'(t) dt
Solution & Explanation:Mass rate equals concentration times volume flow rate, with units milligrams per second. Integrating that rate over time gives total mass.

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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 (Choose the integral model that best fits a described physical, biological, or economic situation. 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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