The Definite Integral | Riemann Sums, Properties, and Area Interpretation

📚 Learn How to Understand and Evaluate Definite Integrals In this video, you’ll learn the definition of the definite integral as the limit of a Riemann sum and how it represents the exact area under a curve. We’ll explore how to evaluate definite integrals using limits, interpret them geometrically as areas, and apply the comparison property to estimate their values. 🔍 What You’ll Learn in This Video: ✅ Riemann Sums ✔ Definition: A Riemann sum is an approximation of the area under a curve over a closed interval [a, b] ✔ Δx = (b – a)/n and Rₙ = Σ f(xᵢ*) Δx ✔ Sample points:  • Left endpoints  xᵢ* = xᵢ₋₁  • Right endpoints xᵢ* = xᵢ  • Midpoints    xᵢ* = (xᵢ₋₁ + xᵢ)/2 ✔ Example: Evaluate the Riemann sum for f(x) = 2x – 4 on [0, 3] using 6 subintervals with  (a) Right endpoints (b) Left endpoints ✅ Definition of the Definite Integral ✔ ∫ₐᵇ f(x) dx = lim Σ f(xᵢ*) Δx as n → ∞ ✔ The definite integral of a continuous function equals the signed area under f(x) from a to b ✔ Parts of the notation:  ∫ = integral sign  f(x) = integrand  dx = differential  a = lower limit   b = upper limit ✅ Properties of Definite Integrals ✔ ∫ₐᵃ f(x) dx = 0 ✔ ∫ₐᵇ f(x) dx = –∫ᵇₐ f(x) dx ✔ ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx (a ≤ c ≤ b) ✔ Example: Express lim Σ ((xᵢ*)² + 4) Δx on [2, 5] as a definite integral ✅ Evaluating Definite Integrals Using Limits ✔ Use the limit definition to evaluate ∫₁³ (2x – x³) dx ✔ Apply formulas for Σi, Σi², Σi³ to simplify ✔ Compute the limit to find the exact value ✅ Evaluating Definite Integrals as Areas ✔ Interpret the definite integral as total area (above minus below the x-axis) ✔ Example 1: ∫₀¹⁰ |x – 5| dx ✔ Example 2: ∫₋₅⁰ (1 + √(25 – x²)) dx ✅ Comparison Property for Definite Integrals ✔ If m ≤ f(x) ≤ M on [a, b], then m(b – a) ≤ ∫ₐᵇ f(x) dx ≤ M(b – a) ✔ Used to estimate the integral’s value by bounding it above and below ✔ Example 1: ∫₀¹ x³ dx ✔ Example 2: ∫₀² x e⁻ˣ dx By the end of this video, you’ll understand the definition, properties, and geometric meaning of definite integrals and how to estimate or evaluate them using limits and comparison. 📥 Download Free Guided Notes! 👉 https://www.understandthemath.com/cal... 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Connect & Support TikTok:   / understandthemath   Instagram:   / understandthemath   Facebook: https://www.facebook.com/profile.php?... Support: https://buymeacoffee.com/understandth... Timestamps 00:00 Introduction 00:53 Riemann Sum Definition and Example 12:22 Definite Integral Definition, Properties, and Example 16:32 Example: Evaluating a Definite Integral Using Limits 31:24 Examples: Evaluating Definite Integrals as Areas 38:24 Comparison Property for Definite Integrals and Examples More About Understand The Math Understand The Math offers clear, accessible, and engaging mathematics instruction for students and lifelong learners. I'm Cheryl Hile, the founder of Understand The Math, with a Ph.D. in Engineering Science and Applied Mathematics and over 25 years of university teaching experience. My goal is to provide accurate, high-quality math instruction through step-by-step explanations and carefully worked-out examples. Each video includes free guided notes, linked in the description. These notes help students follow along, highlight key concepts, and practice problems, making review easier and more effective. #Calculus #Calculus1 #DefiniteIntegrals #RiemannSums #Integration #MathHelp #StewartCalculus #UnderstandTheMath