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/1nyZAxMFIv3phTs8a9uQiZ7WKvrQ4fBKt/view?usp=drive_link
• Answers: https://drive.google.com/file/d/16wbbNzLH-O0LGu6SEvM5Nq9gNQlpoR-H/view?usp=drive_link

💡Chapters:
00:00 Definite integral, definition
01:38 Integral to Riemann sum
03:33 Worked example

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