4: Introduction to Integrals
- Page ID
- 188720
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While derivatives measure instantaneous rates of change, integrals allow us to measure accumulation. Integrals provide a mathematical framework to compute areas, total change, and accumulated quantities.
The concept begins with approximating area using rectangles, formalized through Riemann sums. From there, we define the definite integral and explore its properties. The Fundamental Theorem of Calculus reveals the deep connection between differentiation and integration, providing a powerful tool to evaluate integrals. Finally, we introduce the Substitution Rule, a method that simplifies integrals by reversing the Chain Rule.
Integrals are essential in mathematics, physics, engineering, and economics, where they represent accumulated quantities such as distance traveled, total growth, and area under a curve.
Topics in this Chapter
- Riemann Sums
- Approximate area under a curve using sums of rectangles.
- Understand left, right, and midpoint Riemann sums.
- Connect the idea of approximation to the exact value of an integral.
- The Definite Integral
- Define the definite integral as the limit of Riemann sums.
- Interpret the definite integral as the net area under a curve.
- Learn the properties of the definite integral (linearity, additivity, etc.).
- The Fundamental Theorem of Calculus
- State and apply both parts of the Fundamental Theorem of Calculus.
- Connect differentiation and integration as inverse processes.
- Evaluate definite integrals efficiently using antiderivatives.
- The Substitution Rule
- Apply substitution to compute integrals, reversing the Chain Rule.
- Solve integrals involving composite functions.
- Practice substitution in both indefinite and definite integrals.
Why This Matters
Integration is a central idea of calculus, balancing the concept of differentiation. It allows us to:
- Compute exact areas, accumulated quantities, and total change.
- Solve real-world problems in physics, engineering, biology, and economics.
- Establish the foundation for advanced applications such as volumes, work, probability, and differential equations.
Integrals transform local rates of change into global accumulation, providing one of the most powerful tools in mathematics.