Understanding properties of definite integrals:

• The Fundamental Theorem of Calculus connects integration and differentiation.


  True
• What does ∫ₐᵇ f(x) dx represent?

  Definite integral
• ∫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.


  True
• 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 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.


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

  True
• ∫ₐᵇ [c₁f₁(x) + c₂f₂(x)] dx = c₁∫ₐᵇ f₁(x) dx + c₂∫ₐᵇ f₂(x) dx describes the linearity
• What condition must be satisfied for the Summation Property to apply?

  a < c < b
• The Summation Property allows dividing a complex integral into simpler subintervals for easier evaluation.


  True
• ∫ₐᵇ f(x) dx = -∫ᵇₐ f(x) dx expresses the property of order of integration
• What does a definite integral calculate?

  Signed area
• The Fundamental Theorem of Calculus states that ∫ₐᵇ f(x) dx = F(b) - F(a)
• What does the linearity property allow in definite integrals?

  Separating sums
• The linearity property simplifies definite integrals by multiplying each function by its constant
• In the linearity property, c₁ and c₂ must be constants.


  True
• Steps to simplify definite integrals using the linearity property

  1️⃣ Multiply each function by its constant

  2️⃣ Integrate each function separately

  3️⃣ Add the results
• In the constant multiple property, 'c' must be a constant.


  True
• Steps to simplify definite integrals using the constant multiple property

  1️⃣ Factor out the constant

  2️⃣ Integrate the remaining function

  3️⃣ Multiply the result by the constant
• In the addition property, a < c < b must hold true.


  True
• Steps to simplify definite integrals using the summation property

  1️⃣ Split the integral into subranges

  2️⃣ Integrate each subrange separately

  3️⃣ Add the results
• ∫ₐᵇ f(x) dx is equal to -∫ᵇₐ f(x) dx.


  True
• What does changing the order of integration from ∫ₐᵇ f(x) dx to ∫ᵇₐ f(x) dx do to the integral?

  Changes the sign
• The order of integration property simplifies when boundaries need switching
• ∫₂₀ x² dx equals -8/3


  True
• Match the definite integral property with its formula:

  Order of Integration ↔️ ∫ₐᵇ f(x) dx = -∫ᵇₐ f(x) dx

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

  Constant Multiple ↔️ ∫ₐᵇ c·f(x) dx = c·∫ₐᵇ f(x) dx

  Summation ↔️ ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx
• When integrating from a point to itself, the area under the curve is zero


  True
• What is the formula for the Symmetry Property of definite integrals when f(x) is an even function?

  ∫₋ₐᵃ f(x) dx = 2∫₀ᵃ f(x) dx
• The Symmetry Property simplifies the evaluation of definite integrals for even functions by integrating over the entire interval [-a, a].

  False
• Arrange the following properties of definite integrals in order of their complexity and simplification ability.

  1️⃣ Linearity

  2️⃣ Constant Multiple

  3️⃣ Symmetry Property
• The Substitution Property simplifies definite integrals by transforming them into simpler forms through variable changes.

  True
• The Substitution Property requires changing the limits of integration when transforming the variable.


  True
• A definite integral is denoted as ∫ₐᵇ f(x) dx
• To find the area under the curve of f(x) = x² from x = 0 to x = 2, we use the antiderivative F(x) = x³/3.

  True