8.8 Volumes with Cross Sections: Triangles and Semicircles

• The formula for volume using cross sections is Volume = ∫ A(x) dx


  True
• The area formula for a triangle is A(x) = 1/2 * b(x) * h(x)
• In the area formulas, b(x) and h(x) represent the base and height of a triangle, respectively.

  True
• Triangular cross sections have an area given by the formula A(x) = 1/2 * b(x) * h(x)
• An example of setting up the integral is Volume = ∫₀² (1/2) * (x² + x) dx
• The area function for a triangle is given by A(x) = 1/2 * b(x) * h(x)
• The limits of integration for the volume calculation are determined by the range of x values where the solid exists.
• The volume integral for b(x) = x + 1 and h(x) = x, from x = 0 to x = 2, is ∫₀² (1/2) * (x² + x) dx


  True
• The volume calculation for b(x) = x + 1 and h(x) = x, from x = 0 to x = 2, involves integrating 1/2 * (x² + x) dx


  True
• The volume formula for an object with triangular cross sections is Volume = ∫ 1/2 * b(x) * h(x) dx
• The volume of an object with triangular cross sections is found by integrating the area function over the relevant range of x
• What is the calculated volume in cubic units for triangular cross sections with b(x) = x + 1 and h(x) = x, from x = 0 to x = 2?

  7/3
• The area formula for a semicircle is A(x) = π/4 * d(x)², where d(x) is the diameter.


  True
• What are the three main steps involved in setting up an integral for triangular cross sections?

  Identify area function, determine limits, set up integral
• Arrange the steps for setting up an integral for triangular cross sections in the correct order:

  1️⃣ Identify the area function A(x)

  2️⃣ Determine the limits of integration

  3️⃣ Set up the integral
• The limits of integration for triangular cross sections define the range of x values where the solid exists.
• The area function for triangular cross sections is A(x) = 1/2 * b(x) * h(x).


  True
• Arrange the steps for calculating the volume of triangular cross sections in the correct order:

  1️⃣ Identify the area function

  2️⃣ Set up the volume integral

  3️⃣ Evaluate the integral
• The volume of an object with semicircular cross sections is found by integrating the area function ∫ (π/4) * d(x)² dx over the relevant range of x
• If the diameter of the semicircular cross sections is d(x) = 2x, what is the volume of the object from x = 0 to x = 3?

  13.5π cubic units
• What is the formula for the area of a semicircular cross section at position x?

  A(x) = π/4 * d(x)²
• Steps to set up the integral for an object with semicircular cross sections

  1️⃣ Identify the area function A(x) = π/4 * d(x)^2

  2️⃣ Set up the volume integral: Volume = ∫ A(x) dx
• What is the general volume formula for an object with semicircular cross sections?

  Volume = ∫ A(x) dx
• If d(x) = 2x and the limits are from x = 0 to x = 3, the volume integral is ∫₀³ (π/4) * (2x)^2 dx
• What is the volume of an object with semicircular cross sections where d(x) = 2x and the limits are from x = 0 to x = 3?

  13.5π cubic units
• To calculate the volume of an object with semicircular cross sections, the first step is to identify the diameter function
• The area function for semicircular cross sections is A(x) = π/4 * d(x)^2

  True
• Match the cross section shape with its area formula

  Triangle ↔️ A(x) = 1/2 * b(x) * h(x)

  Semicircle ↔️ A(x) = π/4 * d(x)^2
• What is the volume integral for an object with triangular cross sections where b(x) = x + 1 and h(x) = x, from x = 0 to x = 2?

  Volume = ∫₀² (1/2) * (x² + x) dx
• The volume formula for triangular cross sections is Volume = ∫ (1/2) * b(x) * h(x) dx


  True
• What is the formula for the area of a triangular cross section?

  A(x) = 1/2 * b(x) * h(x)
• What is the volume formula for triangular cross sections?

  Volume = ∫ (1/2) * b(x) * h(x) dx
• The area of a triangular cross section is given by the formula A(x) = 1/2 * b(x) * h(x).


  True
• What is the first step in setting up an integral for triangular cross sections?

  Identify the area function
• The area function for triangular cross sections is A(x) = 1/2 * b(x) * h(x).


  True
• What is the first step in calculating the volume of an object with triangular cross sections?

  Identify the area function
• If the base is b(x) = x + 1 and the height is h(x) = x, the volume from x = 0 to x = 2 is 7/3 cubic units.
• What is the volume formula for semicircular cross sections?

  Volume = ∫ (π/4) * d(x)^2 dx
• Match the cross section shape with its volume formula:

  Triangle ↔️ ∫ (1/2) * b(x) * h(x) dx

  Semicircle ↔️ ∫ (π/4) * d(x)^2 dx
• The general formula for calculating volume using integration is Volume = ∫ A(x) dx.