MATH 1150
Calculus of Functions of One Variable II
Complete Study Guide — All 13 Chapters
Independent Study Resource
Not affiliated with or endorsed by Yale University
Aligned with the Yale MATH 1150 syllabus
Edition 2026
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,Table of Contents
1. Riemann Sums and the Definite Integral
2. The Fundamental Theorem of Calculus and Net Change
3. Integration by Substitution
4. Integration by Parts
5. Trigonometric Integrals and Trigonometric Substitution
6. Partial Fractions and Improper Integrals
7. Applications of Integration: Area, Volume, and Arc Length
8. Sequences and Infinite Series
9. Convergence Tests
10. Power Series and Taylor Series
11. Parametric Equations
12. Polar Coordinates
13. Differential Equations and Integration Review
14. Glossary of Key Terms
15. Formula Sheet
16. Final Exam Strategy and Study Checklist
17. References and Attribution
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, Chapter 1 — Riemann Sums and the Definite
Integral
Learning Objectives
1. Define the definite integral as a limit of Riemann sums.
2. Compute left, right, and midpoint Riemann sums.
3. Interpret the definite integral as net signed area.
4. Apply properties of the definite integral.
Key Definitions
Term Definition
Riemann Sum An approximation of the area under a curve using rectangles: Σ f(xᵢ) Δx.
Definite Integral ∫ab f(x) dx = limn→∞ Σ f(xᵢ) Δx.
Net Signed Area Area above the x-axis minus area below.
Partition A division of an interval into subintervals.
1.1 Riemann Sums
Δx = (b − a) / n
Left sum: Σ f(xᵢ₋₁) Δx
Right sum: Σ f(xᵢ) Δx
Midpoint sum: Σ f((xᵢ₋₁ + xᵢ)/2) Δx
1.2 Properties of Definite Integrals
∫ab c·f(x) dx = c ∫ab f(x) dx
∫ab [f(x) ± g(x)] dx = ∫ab f(x) dx ± ∫ab g(x) dx
∫aa f(x) dx = 0
∫ab f(x) dx = −∫ba f(x) dx
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Calculus of Functions of One Variable II
Complete Study Guide — All 13 Chapters
Independent Study Resource
Not affiliated with or endorsed by Yale University
Aligned with the Yale MATH 1150 syllabus
Edition 2026
Explore our developer-friendly HTML to PDF API Printed using PDFCrowd HTML to PDF
,Table of Contents
1. Riemann Sums and the Definite Integral
2. The Fundamental Theorem of Calculus and Net Change
3. Integration by Substitution
4. Integration by Parts
5. Trigonometric Integrals and Trigonometric Substitution
6. Partial Fractions and Improper Integrals
7. Applications of Integration: Area, Volume, and Arc Length
8. Sequences and Infinite Series
9. Convergence Tests
10. Power Series and Taylor Series
11. Parametric Equations
12. Polar Coordinates
13. Differential Equations and Integration Review
14. Glossary of Key Terms
15. Formula Sheet
16. Final Exam Strategy and Study Checklist
17. References and Attribution
Explore our developer-friendly HTML to PDF API Printed using PDFCrowd HTML to PDF
, Chapter 1 — Riemann Sums and the Definite
Integral
Learning Objectives
1. Define the definite integral as a limit of Riemann sums.
2. Compute left, right, and midpoint Riemann sums.
3. Interpret the definite integral as net signed area.
4. Apply properties of the definite integral.
Key Definitions
Term Definition
Riemann Sum An approximation of the area under a curve using rectangles: Σ f(xᵢ) Δx.
Definite Integral ∫ab f(x) dx = limn→∞ Σ f(xᵢ) Δx.
Net Signed Area Area above the x-axis minus area below.
Partition A division of an interval into subintervals.
1.1 Riemann Sums
Δx = (b − a) / n
Left sum: Σ f(xᵢ₋₁) Δx
Right sum: Σ f(xᵢ) Δx
Midpoint sum: Σ f((xᵢ₋₁ + xᵢ)/2) Δx
1.2 Properties of Definite Integrals
∫ab c·f(x) dx = c ∫ab f(x) dx
∫ab [f(x) ± g(x)] dx = ∫ab f(x) dx ± ∫ab g(x) dx
∫aa f(x) dx = 0
∫ab f(x) dx = −∫ba f(x) dx
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