Understanding properties of definite integrals:

• The Fundamental Theorem of Calculus connects integration and differentiation.


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• What does ∫ₐᵇ f(x) dx represent?

  Definite integral
• A definite integral represents the signed area under the curve
• ∫ₐᵇ f(x) dx = F(b) - F(a), where F(x) is the antiderivative
• The linearity property allows us to break down complex integrals.


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• What does the Constant Multiple Property allow us to do with definite integrals?

  Factor out constants
• The Linearity Property breaks down complex integrals, while the Constant Multiple Property simplifies by factoring out constants
• The Constant Multiple Property allows us to factor out constants from the integrand to simplify definite integrals.


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• Match the definite integral property with its formula:

  Linearity Property ↔️ ∫ₐᵇ [c₁f₁(x) + c₂f₂(x)] dx = c₁∫ₐᵇ f₁(x) dx + c₂∫ₐᵇ f₂(x) dx

  Constant Multiple Property ↔️ ∫ₐᵇ c·f(x) dx = c·∫ₐᵇ f(x) dx
• In a definite integral, the integrand is denoted by f(x).


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• The Fundamental Theorem of Calculus states that ∫ₐᵇ f(x) dx equals F(b) - F(a)
• The Linearity Property states that ∫ₐᵇ [c₁f₁(x) + c₂f₂(x)] dx equals c₁∫ₐᵇ f₁(x) dx + c₂∫ₐᵇ f₂(x) dx.
• What does the Addition Property of definite integrals allow us to do?

  Divide the interval into subintervals
• The Summation Property of definite integrals is equivalent to the Addition Property.


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• The addition property simplifies integrals by dividing the interval of integration into subintervals.

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• ∫ₐᵇ [c₁f₁(x) + c₂f₂(x)] dx = c₁∫ₐᵇ f₁(x) dx + c₂∫ₐᵇ f₂(x) dx describes the linearity
• The Summation Property states that ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx if a < c < b.

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• The Fundamental Theorem of Calculus states that ∫ₐᵇ f(x) dx = F(b) - F(a)
• The Fundamental Theorem of Calculus relates definite integrals to antiderivatives.

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• What does the linearity property allow in definite integrals?

  Separating sums
• The linearity property states that ∫ₐᵇ [c₁f₁(x) + c₂f₂(x)] dx = c₁∫ₐᵇ f₁(x) dx + c₂∫ₐᵇ f₂(x) dx.


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• The linearity property simplifies definite integrals by multiplying each function by its constant
• The linearity property allows the integral of a sum of functions multiplied by constants to be separated into individual integrals
• In the linearity property, c₁ and c₂ must be constants.


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• ∫ₐᵇ [c₁f₁(x) + c₂f₂(x)] dx = c₁∫ₐᵇ f₁(x) dx + c₂∫ₐᵇ f₂(x) dx, where c₁ and c₂ are constants
• The Constant Multiple Property states ∫ₐᵇ c·f(x) dx = c·∫ₐᵇ f(x) dx.


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• The Constant Multiple Property states that ∫ₐᵇ c·f(x) dx equals c·∫ₐᵇ f(x)
• The Linearity Property allows us to break down complex integrals into simpler ones.
• What does a definite integral calculate?

  Signed area under a curve
• Order the steps to evaluate a definite integral using the Fundamental Theorem of Calculus:

  1️⃣ Find the antiderivative F(x) of f(x)

  2️⃣ Evaluate F(b) - F(a)
• Match the component of a definite integral with its description:

  ∫ ↔️ Integral symbol

  a ↔️ Lower limit of integration

  b ↔️ Upper limit of integration

  f(x) ↔️ Integrand
• The Linearity Property allows us to integrate each function separately and then add the results.


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• The Addition Property states that ∫ₐᵇ f(x) dx equals ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x)
• ∫₀² x² dx = ∫₀¹ x² dx + ∫₁² x² dx illustrates the addition
• What condition must be satisfied for the Summation Property to apply?

  a < c < b
• ∫ₐᵇ c·f(x) dx = c·∫ₐᵇ f(x) dx illustrates the constant multiple
• The Summation Property allows dividing a complex integral into simpler subintervals for easier evaluation.


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• What changes when the order of integration is reversed?

  The sign
• ∫ₐᵇ f(x) dx = -∫ᵇₐ f(x) dx expresses the property of order of integration
• Reversing the order of integration changes the sign of the definite integral.


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