6.5 Interpreting the Behavior of Accumulation Functions Involving Area

• The rate function in an accumulation function describes how a quantity changes over time
• The accumulation function F(x) represents the total change in a quantity from a starting point 'a' to a current point 'x'.


  True
• Over the interval [0, 2], the area under the curve of f(x) = 2x is 4
• The area under a curve is calculated using definite integrals.

  True
• The value of the definite integral represents the area
• What is the area under f(x) = x² from x = 0 to x = 2?

  8/3
• Accumulation functions are calculated by integrating a rate function.
• What does the rate function f(x) describe?

  Change of quantity over time
• The area under a curve is calculated using definite integrals.


  True
• What is the area under f(x) = x² from x = 0 to x = 2?

  8/3
• The integral of the rate function f(x) over [a, x] gives the accumulation function F(x).


  True
• The derivative of an accumulation function F(x) is the original rate function f(x)
• The accumulation function F(x) represents the area under the curve of the rate function f(x) from the starting point a
• Zero net change occurs when the positive and negative areas under a rate function cancel each other out.


  True
• The accumulation function F(x) represents the total change in a quantity from a starting point 'a' to a current point 'x'.

  True
• If f(x) = 2x represents the rate of water flow into a tank, then F(x) = ∫₀ˣ 2t dt calculates the total amount of water added from time 0 to time x.


  True
• Steps to calculate and interpret accumulation functions

  1️⃣ Integrate the rate function f(x) over the interval [a, x]

  2️⃣ Find the area under the curve

  3️⃣ Use the accumulation function F(x) to find the total change from 'a' to 'x'
• What does the accumulation function F(x) represent if f(x) = 2x describes the rate of water flow into a tank?

  Total amount of water added
• The integral of a rate function f(x) over the interval [a, x] gives the area under the curve.


  True
• What is the purpose of definite integrals in calculating the area under a curve?

  Calculate the exact area
• Evaluating the definite integral involves finding the antiderivative and subtracting the values at the limits of integration.


  True
• Match the property of accumulation functions with its formula:

  Derivative ↔️ F'(x) = f(x)

  Integral ↔️ F(x) = ∫ f(x) dx
• Zero net change occurs when the positive and negative areas under a curve cancel each other out.


  True
• What is the net change under the curve of f(x) = x - 2 from x = 0 to x = 4?

  0
• A zero net change in accumulated area means there is no overall change
• What is the total area under f(x) = x - 2 from x = 0 to x = 4?

  0
• The accumulation function F(x) represents the total change in quantity from a starting point to the current point x.


  True
• How is the change in quantity calculated using the accumulation function F(x)?

  F(b) - F(a)
• If F'(x) > 0, the accumulation function F(x) is increasing.


  True
• The Fundamental Theorem of Calculus states that if F(x) is the antiderivative of f(x), then \int_{a}^{b} f(x) \, dx = F(b) - F(a)</latex> is True
• If F(x) is the accumulation function of f(x), the area under the curve of f(x) from a to b is equal to F(b) - F(a).


  True
• To evaluate a definite integral using the Fundamental Theorem, one must first find the antiderivative
• What do accumulation functions represent in calculus?

  Total change in a quantity
• What does the integral in an accumulation function calculate?

  Area under the curve
• What does the accumulation function F(x) = ∫₀ˣ 2t dt calculate if f(x) = 2x represents the rate of water flow into a tank?

  Total amount of water added
• Steps for using an accumulation function

  1️⃣ Integrate the rate function f(x) over [a, x]

  2️⃣ Find the area under the curve

  3️⃣ Calculate the net change in the quantity

  4️⃣ Use F(x) to track cumulative change
• What is the next step after setting up the definite integral to find the area under a curve?

  Find the antiderivative
• Match the action with the corresponding formula for finding the area under a curve:

  Set up integral ↔️ ∫ₐᵇ f(x) dx

  Find antiderivative ↔️ F(x) = ∫ f(x) dx

  Evaluate integral ↔️ F(b) - F(a)

  Interpret result ↔️ Area under the curve
• What does the accumulation function F(x) represent?

  Total change from 'a' to 'x'
• The integral of f(x) over [a, x] represents the net change in the quantity.


  True