Setting up integrals to find the area between two curves:

• When dealing with implicit equations, try to express both equations in terms of either x or y
• When calculating the area between two curves, the first step is to identify the functions
• If the curves are defined as x = f(y) and x = g(y), the equation to find intersection points is f(y) = g(y)


  True
• Steps to find the area between two curves

  1️⃣ Identify the functions defining the curves

  2️⃣ Find the points of intersection

  3️⃣ Determine the integrand

  4️⃣ Establish the limits of integration

  5️⃣ Write the definite integral

  6️⃣ Evaluate the integral
• When the curves are defined as functions of y, they take the form x = g(y)
• Setting the two function equations equal to each other is the first step in finding their intersection points


  True
• What is the integrand when finding the area between y = x^2 and y = x + 2 from x = -1 to x = 2?

  (x + 2) - x^2
• The area between y = x^2 and y = x + 2 is calculated from x = -1 to x = 2
• The integrand for the area between y = x^2 and y = x + 2 is (x + 2) - x^2
• For y = f(x), the top function has higher y-values
• To ensure the area is always positive, use absolute value
• Match the case with its example for determining integration limits:

  Known Intersection Points ↔️ y = x^2 and y = x + 2 intersect at x = -1 and x = 2

  No Explicit Intersection Points ↔️ Find the area between y = x^2 and y = x + 2 from x = 0 to x = 1

  Bounded Region Only ↔️ Curves x = y^2 and x = 2y intersect at y = 0 and y = 2
• Curves defined as x = g(y) are used when they are more easily described in terms of y.


  True
• The intersection points for y = x^2 and y = x + 2 are x = 2 and x = -1.


  True
• Steps to determine the integrand when finding the area between two curves:

  1️⃣ Graph the functions to visualize the top/bottom or left/right function

  2️⃣ Subtract the bottom function from the top function for y = f(x)

  3️⃣ Subtract the right function from the left function for x = g(y)

  4️⃣ Use absolute value if the order changes within the interval
• The form y = f(x) is the most common way to define curves when finding the area between them

  True
• When is the form x = g(y) used to define curves?

  When easier in terms of y
• What equation should be set up if the curves are in the form y = f(x) and y = g(x)?

  f(x) = g(x)
• To find the x or y values of intersection points, you need to solve the equation
• What should you do if the curves are provided as implicit equations?

  Express in terms of x or y
• Graphing the functions is essential to visualize which is the top/bottom or left/right function


  True
• Match the function form with its characteristic:

  y = f(x) ↔️ Vertical difference

  x = g(y) ↔️ Horizontal difference
• For x = g(y), the function with greater x-values is considered the right function.


  True
• The integrand for y = f(x) is based on vertical difference and integration with respect to x.

  True
• If curves intersect at x = -1 and x = 2, the limits of integration are -1 to 2.

  True
• Steps to find the points of intersection between two curves:

  1️⃣ Set the two function equations equal to each other

  2️⃣ Solve the equation to find the x or y values

  3️⃣ Plug the values into either original function
• What is the form of the function for which we use the left function as the higher value?

  x = g(y)
• In the example y = x^2 and y = x + 2, what is the top function?

  y = x + 2
• When using y = f(x), the integrand is based on vertical difference, and integration is with respect to x.

  True
• Match the case with its explanation for establishing the limits of integration:

  Known Intersection Points ↔️ Limits are x or y values of intersection

  No Explicit Intersection Points ↔️ Use the given interval as limits

  Bounded Region Only ↔️ Use intersection points as limits
• The integrand for the area between y = x^2 and y = x + 2 is (x + 2) - x^2
• Identifying the functions defining the curves is the first step in finding the area between them

  True
• To find the points of intersection between two curves, set their equations equal to each other
• What should you do after finding the x or y values of the intersection points?

  Plug into original function
• The form y = f(x) is the most common way to define curves when finding the area between them


  True
• After finding the x-values of intersection points, plug them back into either original function
• Match the function form with its integration characteristic:

  y = f(x) ↔️ Integration with respect to x

  x = g(y) ↔️ Integration with respect to y

  Vertical difference ↔️ y-values are subtracted

  Horizontal difference ↔️ x-values are subtracted
• The top function between y = x^2 and y = x + 2 is y = x + 2.

  True
• The integrand is the function that is integrated to calculate the area between two curves.


  True
• Steps to determine the integrand for the area between two curves:

  1️⃣ Graph the functions

  2️⃣ Subtract the bottom from the top for y = f(x) or the right from the left for x = g(y)

  3️⃣ Use absolute value if the order changes