Setting up integrals to find the area between two curves:

• The top curve is the one with larger y-values at each point in the interval.


  True
• What is the graphical inspection method for identifying top and bottom curves?

  Plot curves and visually determine
• Consider $f(x) =$$x^{2}$ and $g(x) =$$x +$$2$ from $x =$$- 1$ to $x =$$2$. Which curve is on top?


  $g(x)$
• The intersection points represent the values of x where the top and bottom curves cross.
• What must you identify to find the area between two curves?

  Top and bottom curves
• How do you find the intersection points of two curves?

  Set their equations equal
• What do the intersection points define in the region between two curves?

  The beginning and end
• What are the limits of integration for the area between f(x) = x^2 and g(x) = x + 2?

  [-1, 2]
• When calculating the area between two curves, it is crucial to identify the top and bottom curves
• Methods to identify the top and bottom curves

  1️⃣ Graphical Inspection

  2️⃣ Algebraic Comparison

  3️⃣ Test Value
• The test value method involves evaluating both functions at a single x-value within the interval.


  True
• To find the intersection points of two curves, you must set their equations equal to each other.


  True
• In the area integral, $f(x)$ represents the top curve and $g(x)$ represents the bottom curve.


  True
• The area formula between two curves is ∫[a, b] [f(x) - g(x)] dx, where f(x) is the top curve and g(x) is the bottom curve.

  True
• Steps to find the intersection points of two curves

  1️⃣ Set the equations of the top and bottom curves equal to each other

  2️⃣ Solve the resulting equation to find the x-coordinate(s)

  3️⃣ Evaluate the y-coordinates by plugging the x-coordinate(s) into either curve equation
• The limits of integration can be directly provided in the problem or found from the intersection points of the curves.


  True
• What does the integral ∫[a, b] [f(x) - g(x)] dx calculate?

  Area between two curves
• What is the first step in calculating the area between two curves using integration?

  Identify top and bottom curves
• The bottom curve is the one with smaller y-values at each point in the interval.


  True
• In the algebraic comparison method, we choose several x-values within the interval.
• What is the formula for the area between two curves?

  $\int_{a}^{b} [f(x) - g(x)] dx$
• Which curve has higher y-values at each point within the interval?

  Top Curve
• The top curve is defined as the curve with higher y-values
• To find the intersection points, you solve the equation resulting from setting the curve equations equal to find the x-coordinate
• If the interval is not given, the limits of integration are found from the intersection
• The area between two curves is calculated using the integral formula ∫[a, b] [f(x) - g(x)] dx