Definite Integral, Definition from Riemann sum, Formula, Symbol, Example - Calculus

A definite integral calculates the net signed area between a function's curve and the x-axis over a specific interval, defined by a lower limit (a) and an upper limit (b) of integration. The definite integral of a function f(x) from a to b, written as ∫ᵇₐ f(x) dx, is found by calculating the antiderivative F(x) of f(x) and then evaluating F(b) - F(a). The result is a single number representing the exact area, which can be positive, negative, or zero depending on whether the area is above, below, or crosses the x-axis. 💡Key Components and Notation • Integral Symbol (∫): Indicates the operation of integration. • Integrand (f(x)): The function being integrated. • Limits of Integration (a and b): The lower (a) and upper (b) bounds of the interval over which the area is calculated. • Differential (dx): Indicates the variable of integration. 💡How to Evaluate a Definite Integral • Find the Antiderivative: Determine the indefinite integral (antiderivative) F(x) of the function f(x). • Substitute Limits: Substitute the upper limit (b) and the lower limit (a) into the antiderivative F(x). • Subtract: Subtract the result from the lower limit substitution from the result of the upper limit substitution: F(b) - F(a). 💡Example To find the definite integral of 2x from 1 to 2: • Antiderivative: The antiderivative of 2x is x². • Evaluate at Limits: ◦ Upper limit: (2)² = 4 ◦ Lower limit: (1)² = 1 • Subtract: 4 - 1 = 3. • Therefore, ∫²₁ 2x dx = 3. 💡When and Why to Use Definite Integrals Definite integrals are used to: • Calculate Exact Areas: Find the precise area under a curve over a given interval. • Measure Net Signed Area: Determine the total area above the x-axis minus the total area below the x-axis within the interval. • Model Accumulation: Measure the total buildup of a quantity over a period when its rate of change is known, such as the total distance traveled by a car given its velocity function over time. 💡Worksheets are provided in PDF format to further improve your understanding: • Questions Worksheet: https://drive.google.com/file/d/1nyZA... • Answers: https://drive.google.com/file/d/16wbb... 💡Chapters: 00:00 Definite integral, definition 01:38 Integral to Riemann sum 03:33 Worked example 🔔Don’t forget to Like, Share & Subscribe for more easy-to-follow Calculus tutorials. 🔔Subscribe:    / @drofeng   _______________________ ⏩Playlist Link:    • Calculus 1 & 2 Full Course | Limits, Deriv...   _______________________ 💥 Follow us on Social Media 💥 🎵TikTok: https://www.tiktok.com/@drofeng?lang=en 𝕏: https://x.com/DrOfEng 🥊: https://rumble.com/user/drofeng