8.8 Using Integration to Solve Problems Involving Area, Volume, and Accumulation

• What is the Fundamental Theorem of Calculus used to relate?

  Integration and differentiation
• If a function `f(x)` is continuous on an interval `[a, b]`, then the definite integral `∫(a to b) f(x) dx` equals the antiderivative F(b) - F(a)
• The Fundamental Theorem of Calculus states that differentiation "undoes" integration.

  True
• What does the Disk Method calculate?

  Volume of a solid
• `∫(0 to 2) [π(x^2)^2] dx = π [x^5/5] evaluated from 0 to 2 = π (32/5)
• What is the region being revolved in the example using the Disk Method?

  `y = x^2` and `y = 0`
• If `F(x)` is an antiderivative of `f(x)`, then `f(x) = d/dx [F(x)]`.


  True
• When finding the area between two curves, ensure `f(x) ≥ g(x)` throughout the interval.


  True
• In the Washer Method, `R(x)` represents the outer radius and `r(x)` represents the inner radius.


  True
• The Washer Method is an extension of the Disk Method.
• When using the Disk Method, the axis of revolution must be parallel to the variable of integration.

  False
• Steps to calculate volume using the Shell Method

  1️⃣ Define the region and axis of revolution

  2️⃣ Determine the radius r(x) or r(y)

  3️⃣ Find the height h(x) or h(y)

  4️⃣ Set up and evaluate the integral ∫[2πr(x)h(x)] dx or ∫[2πr(y)h(y)] dy
• The Washer Method requires two different radius functions: an outer radius and an inner radius.

  True
• When is the washer method used to calculate volume?

  Region does not touch axis
• The first step in calculating volume using the washer method is to identify the region and axis of revolution.


  True
• The formula for the disk method is Volume = ∫[π(r(x))^2] dx
• The accumulation function is defined as A(x) = ∫(a to x) f(t) dt
• What is the second part of the Fundamental Theorem of Calculus?

  f(x) = d/dx [F(x)]
• The formula for the Disk Method is Volume = ∫(a to b) π(r(x))^2 dx
• The volume formula for the disk method is ∫(a to b) π(r(x))^2 dx
• The Washer Method is used when the region touches the axis of revolution.

  False
• Match the volume method with its use case:

  Disk Method ↔️ Region touches axis

  Washer Method ↔️ Region does not touch axis
• What does `r` represent in the shell method formula?

  Radius of the shell
• What is the first step in calculating volume using the shell method?

  Define the region
• The integral for the shell method always involves multiplying by 2π


  True
• What is the volume of the solid calculated in the example?

  $8π$
• What does `h` represent in the shell method formula?

  Height of the shell
• What is the formula for the disk method when revolving around the x-axis?

  ∫[π(r(x))^2] dx</latex>
• What is the formula for the disk method when revolving around the x-axis?

  Volume = ∫[π(r(x))^2] dx
• The washer method is used when the region being revolved touches the axis of revolution.

  False
• What is the volume of the solid obtained by revolving the region bounded by `y = x^2` and `y = x` from `x = 0` to `x = 1` around the x-axis?

  $\frac{2π}{15}$
• What is the formula for the accumulation function `A(x)`?

  A(x) = ∫(a to x) f(t) dt
• Steps to calculate volume using the disk or washer method

  1️⃣ Identify the region and axis of revolution

  2️⃣ Determine the appropriate radius function(s)

  3️⃣ Apply the disk or washer method formula

  4️⃣ Evaluate the definite integral to find the volume
• According to the Fundamental Theorem of Calculus, what is `A(x)` if `f(t)` has an antiderivative `F(x)`?

  $F(x) - F(a)$
• The accumulation function `A(x)` is calculated by finding the antiderivative `F(x)` and evaluating `F(x) - F(a)
• The Washer Method calculates volume by subtracting the volume of the inner radius from the outer radius.


  True
• Part 2 of the Fundamental Theorem of Calculus states that differentiation "undoes" integration.

  True
• The area between two curves is calculated using the integral `∫(a to b) [f(x) - g(x)] dx
• If `f(x) = 3x^2`, what is an antiderivative `F(x)` of `f(x)`?

  x^3
• Match the method with its formula for calculating volume:

  Disk Method ↔️ ∫[π(r(x))^2] dx

  Washer Method ↔️ ∫[π((R(x))^2 - (r(x))^2)] dx