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George Brown Polytechnic · MATH2066

George Brown Polytechnic MATH2066 tutoring

Help with derivatives, optimization, integration, areas and volumes.

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Course overview

MATH2066 combines differential and integral calculus in George Brown’s Civil Engineering Technology program. The 2026–27 outline covers derivatives, curve analysis, related rates and optimization, then integration with area and volume applications and integration by parts. These are two connected tools: a derivative can verify an antiderivative, while a definite integral calculates an accumulated quantity.

MATH2066 topics to review

Limits, derivative rules and implicit differentiationTangents, motion, curve sketching and optimizationAntiderivatives, definite integrals, areas and volumesExponential, logarithmic and trigonometric functions; integration by parts

Common difficulties

  • Choosing integration by parts without simplifying the remaining integral
  • Losing the lower-bound contribution

Useful foundations to review

Review the product rule and exponential derivatives before integration by parts. Identify which factor becomes simpler after differentiating and which factor is easy to integrate. Keep the original bounds until the antiderivative is ready, and evaluate both endpoints.

Registration requirements

The guidance above is study preparation, not an eligibility decision. Confirm current prerequisites, restrictions and availability with George Brown Polytechnic.

Check the official course catalogue

Use the product rule in reverse

Evaluate ∫01xex dx\int_0^1 x e^x\,dx. Explain why integration by parts is useful, and verify the antiderivative.

Show the reasoning

  1. Choose factors that simplify

    Take u=xu=x and dv=ex dxdv=e^x\,dx. Then du=dxdu=dx and v=exv=e^x. Differentiating the polynomial lowers its degree while the exponential stays easy to integrate.

  2. Integrate and check by differentiation

    Integration by parts gives ∫xex dx=xex−ex+C\int xe^x\,dx=xe^x-e^x+C. Differentiating (x−1)ex(x-1)e^x gives ex+(x−1)ex=xexe^x+(x-1)e^x=xe^x, as required.

  3. Evaluate both bounds

    [(x−1)ex]01=0−(−1)=1[(x-1)e^x]_0^1=0-(-1)=1. The integrand is nonnegative on this interval, so a positive integral is consistent with the area interpretation.

Try a changed problem

Use the same antiderivative to evaluate ∫02xex dx\int_0^2 xe^x\,dx.

Check your reasoning

[(x−1)ex]02=e2−(−1)=e2+1[(x-1)e^x]_0^2=e^2-(-1)=e^2+1. The lower-bound term is still −1 before subtraction.

How tutoring can help

A tutor can work through a derivative or integral with you from the first step, explaining why each method applies. For applications, you can build the function, choose the domain or integration bounds, and check the result together. If you’ve already tried the problem, the tutor can help you find the step that needs clarification before practising a similar example.

Exam preparation for MATH2066

Mix differential and integral questions from your assessment chapters. Explain the purpose of the calculation before choosing a rule. In integration, differentiate the result as a check and estimate whether the definite value is plausible from the graph.

Use your section’s outline to choose the chapters to review. We work through permitted practice and comparable problems, without completing graded assignments or exams on your behalf.

Independent practice

Free practice and study guides for MATH2066

35 min

How to Choose an Integration Method

Compare substitution, integration by parts, partial fractions and trigonometric substitution, with sign-aware radical choices, worked examples and progressive checks.

14 questions

Calculus I Practice for Limits and Derivatives

Fourteen questions: an eight-question limits-and-derivatives core, two optional optimization problems and four optional L’Hôpital method-choice questions, with hints and explained answers.

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MATH2066 · official course reference