8.2 Finding the Area Between Curves Expressed as Functions of <latex>y</latex>

• What is the interval over which we find the area between $f(y) =$$3y^{2}$ and g(y) = y^{2} + 2</latex>?


  [0, 2]
• The integral of $2y^{2} - 2$ with respect to $y$ is \frac{2}{3}y^{3} - 2y
• What must be identified to find the area between two curves expressed as functions of $y$?


  Upper and lower curves
• Sketching the curves is the first step in identifying the upper and lower curves

  True
• If $f(y)$ yields larger $x$ values than $g(y)$, then $f(y)$ is the upper
• What are the integration limits for the integral A = \int_{0}^{2} (3y^{2} - 1) \, dy</latex>?

  [0, 2]
• When identifying the upper and lower curves, we determine which function is consistently to the right
• When finding the area between two curves expressed as functions of $y$, we integrate with respect to y
• Steps to find the area between $f(y) =$$3y^{2}$ and $g(y) =$$y^{2} +$$2$ over the interval $[0, 2]$


  1️⃣ Determine the right and left functions

  2️⃣ Set up the integral

  3️⃣ Simplify the integrand

  4️⃣ Integrate

  5️⃣ Evaluate the integral
• Match the step with its description for finding the area between curves

  Determine the right and left functions ↔️ Identify which function yields larger $x$ values

  Set up the integral ↔️ Subtract the left function from the right function

  Simplify the integrand ↔️ Combine like terms in the integral

  Evaluate the integral ↔️ Compute the definite integral
• When comparing curves, the function that consistently yields larger $x$ values is considered the "lower" curve.


  False
• Steps to set up the integral for calculating the area between two curves

  1️⃣ Identify the upper and lower curves

  2️⃣ Write the integrand by subtracting the lower curve from the upper curve

  3️⃣ Define the integration limits

  4️⃣ Set up the definite integral
• The integrand for the curves $f(y) =$$4y^{2}$ and $g(y) =$$y^{2} +$$1$ is 3y^{2} - 1
• What is the formula for finding the area between two curves expressed as functions of $y$?


  $A =$$\int_{c}^{d} [f(y) - g(y)] \, dy$
• Match the curves with their correct descriptions:

  f(y) = 3y^{2}</latex> ↔️ Upper curve

  $g(y) =$$y^{2} +$$2$ ↔️ Lower curve
• When finding the area between two curves expressed as functions of $y$, the lower curve is always subtracted from the upper curve


  True
• The formula for the area between curves in terms of $y$ is $A =$$\int_{c}^{d} [f(y) - g(y)] \, dy$.


  True
• To find the area between curves expressed as functions of $y$, we must first identify the upper curve (to the right) and the lower curve (to the left).
• Steps to set up the integral for area calculation in terms of $y$.


  1️⃣ Identify upper and lower curves

  2️⃣ Write the integrand

  3️⃣ Define integration limits

  4️⃣ Write the integral
• To solve the integral $A =$$\int_{0}^{2} (2y^{2} - 2) \, dy$, we need to find its antiderivative.
• For the example curves $x =$$y^{2}$ and $x =$$2y$, the correct bounds of integration are from $0$ to 2.
• What is the definite integral for finding the area between the curves $f(y) =$$3y^{2}$ and $g(y) =$$y^{2} +$$2$ over the interval $[0, 2]$?


  $A =$$\int_{0}^{2} [3y^{2} - (y^{2} +$$2)] \, dy$
• Which curve is identified as the upper curve in the example provided?

  $f(y) =$$3y^{2}$
• Steps to find the area between curves in terms of $y$.


  1️⃣ Identify the upper and lower curves

  2️⃣ Set up the integral

  3️⃣ Evaluate the integral
• What is the first step in identifying the upper and lower curves for area calculation in terms of $y$?


  Sketch the curves
• For $f(y) =$$3y^{2}$ and g(y) = y^{2} + 2</latex>, the integrand is $3y^{2} - (y^{2} +$$2) =$$2y^{2} - 2$.


  True
• Steps to solve the integral for area calculation in terms of $y$.


  1️⃣ Find the antiderivative

  2️⃣ Evaluate at upper limit

  3️⃣ Evaluate at lower limit

  4️⃣ Subtract lower from upper limit
• Evaluating the antiderivative at the upper limit $y =$$2$ for $\int (2y^{2} - 2) \, dy$ gives $\frac{4}{3}$.


  True
• In the formula for area between curves, $f(y)$ represents the right function and $g(y)$ represents the left function.


  True
• The area between the curves $f(y) =$$3y^{2}$ and $g(y) =$$y^{2} +$$2$ over the interval $[0, 2]$ is $\frac{4}{3}$.


  True
• To set up the integral for calculating the area between curves, we subtract the lower curve from the upper curve.
• To find the area between two curves expressed as functions of $y$, it is crucial to identify which curve is "upper" (right) and which is lower
• The area between two curves is calculated by integrating the difference between the upper and lower curves


  True
• Setting up the integral involves subtracting the lower curve from the upper curve


  True
• The area between the curves $f(y) =$$3y^{2}$ and $g(y) =$$y^{2} +$$2$ over the interval $[0, 2]$ is \frac{4}{3}
• When finding the area between two curves expressed as functions of $y$, we integrate the difference between the upper curve and the lower curve.
• What must be determined first to find the area between curves in terms of $y$?


  Upper and lower curves
• In the integral $A =$$\int_{c}^{d} [f(y) - g(y)] \, dy$, f(y)</latex> represents the upper curve and $g(y)$ represents the lower curve.
• What is the integral for calculating the area between $f(y) =$$3y^{2}$ and $g(y) =$$y^{2} +$$2$ over $[0, 2]$?


  $\int_{0}^{2} (2y^{2} - 2) \, dy$
• Why is it essential to use the correct bounds of integration when calculating the area between curves in terms of $y$?


  To ensure accurate area