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

• A definite integral calculates the area under a curve between two limits along the x-axis
• Match the components of a definite integral with their descriptions:

  ∫ ↔️ Symbol for integration

  a and b ↔️ Limits of integration

  f(x) ↔️ Function being integrated

  dx ↔️ Infinitesimal width along x-axis
• Steps to calculate the area bounded by a curve using a definite integral:

  1️⃣ Identify the curve or function f(x)

  2️⃣ Determine the limits of integration a and b

  3️⃣ Set up the definite integral ∫ab f(x) dx

  4️⃣ Evaluate the definite integral
• To calculate the area bounded by a curve, the limits of integration define the range over which the area is calculated along the x-axis
• The limits of integration in a definite integral represent the x-values between which the area is calculated.


  True
• What is the definite integral to find the area bounded by y = x2 between x = 1 and x = 3?

  ∫13 x2 dx
• What is the area bounded by the curve y = x2 between x = 1 and x = 3?

  14 square units
• To calculate the area bounded by a curve, you must determine the limits of integration
• Steps to calculate the area bounded by a curve using a definite integral

  1️⃣ Identify the curve or function f(x)

  2️⃣ Determine the limits of integration a and b

  3️⃣ Set up the definite integral ∫ab f(x) dx

  4️⃣ Evaluate the definite integral
• The disk method uses circular cross-sections to find the volume of a solid revolved around an axis
• A definite integral calculates the area under a curve between two specified limits of integration


  True
• In the definite integral formula, a and b are the lower and upper limits of integration
• The limits of integration in a definite integral define the y-values over which the area is calculated.

  False
• The washer method is used when cross-sections are circles.

  False
• The derivative of the integral of a function returns the original function according to FTC Part 1.


  True
• What does the Fundamental Theorem of Calculus connect?

  Differentiation and integration
• What does a definite integral calculate?

  Area under a curve
• The integral ∫13 x2 dx represents the area under the curve y = x2 between x = 1 and x = 3.

  True
• What is the first step in calculating the area bounded by a curve using a definite integral?

  Identify the curve f(x)
• What mathematical tool is used to calculate the area bounded by a curve?

  Definite integral
• What is the area bounded by the curve y = x^2 between x = 1 and x = 3?

  14 square units
• The volume of the solid obtained by rotating the region between y = x^2 and y = x around the x-axis from x = 0 to x = 1 is 2π/15 cubic units
• What does ∫ab f(x) dx represent in the context of area calculation?

  Area under the curve
• What is the numerical value of the area bounded by y = x^2 between x = 1 and x = 3?

  14
• Steps to calculate the volume of a solid of revolution using the disk method

  1️⃣ Revolve the solid around an axis

  2️⃣ Identify the cross-sectional areas

  3️⃣ Set up the definite integral

  4️⃣ Evaluate the integral
• What is the volume of the solid obtained by rotating the region between y = x^2 and y = x around the x-axis from x = 0 to x = 1?

  2π/15
• Match the key concept with its definition:

  Accumulation ↔️ Total change over a period

  Rate of Change ↔️ Speed at which a quantity changes

  Definite Integral ↔️ ∫ab f(t) dt
• What is the definite integral that represents the area under the curve y = x2 between x = 1 and x = 3?

  ∫13 x2 dx
• The definite integral ∫13 x2 dx represents the area under the curve y = x2 between x = 1 and x = 3.


  True
• Evaluating the definite integral ∫13 x2 dx gives the area as 14 square units.
• Evaluating the definite integral ∫13 x2 dx results in an area of 14 square units.


  True
• Evaluating the definite integral gives the numerical value of the area


  True
• The formula for the disk method is ∫ab π[f(x)]2 dx


  True
• What is the volume of the solid obtained by rotating y = √x around the x-axis from x = 0 to x = 4?

  8π cubic units
• Match the method with its formula:

  Disk Method ↔️ ∫ab π[f(x)]2 dx

  Washer Method ↔️ ∫ab π([R(x)]2 - [r(x)]2) dx
• In the definite integral, dx represents an infinitesimally small width along the x
• In the definite integral, the lower limit of integration is denoted by a
• What is the volume of the solid obtained by rotating y = √x around the x-axis from x = 0 to x = 4?

  8π
• The total distance traveled by a particle with velocity v(t) = 3t^2 from t = 0 to t = 5 is 125 meters.
• The definite integral of a function is the difference between its antiderivative at the limits of integration.