2.1 Areas between Curves

Calculating the area between curves is a key application of integration. By subtracting one function from another and integrating, we can find the space enclosed by two curves. This technique works for both x and y orientations.

Sometimes curves intersect multiple times, creating compound regions. In these cases, we break the area into smaller parts, integrate each separately, and add the results. Choosing the right variable to integrate with respect to is crucial for simplifying calculations.

Area Between Curves

Area between two curves

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• Find area between curves $[y=f(x)](https://www.fiveableKeyTerm:y=f(x))$ and $[y=g(x)](https://www.fiveableKeyTerm:y=g(x))$ by integrating difference of upper and lower functions with respect to x
  • Solve $f(x)=g(x)$ to determine points of intersection (x-coordinates $a$ and $b$)
  • Integrate $\int_{a}^{b} [f(x)-g(x)] [dx](https://www.fiveableKeyTerm:dx)$ where $f(x)$ is upper function and $g(x)$ is lower function on interval $[a,b]$ (this is an example of a definite integral)
• Find area between curves $[x=h(y)](https://www.fiveableKeyTerm:x=h(y))$ and $[x=k(y)](https://www.fiveableKeyTerm:x=k(y))$ by integrating difference of right and left functions with respect to y
  • Solve $h(y)=k(y)$ to determine points of intersection (y-coordinates $c$ and $d$)
  • Integrate $\int_{c}^{d} [k(y)-h(y)] [dy](https://www.fiveableKeyTerm:dy)$ where $k(y)$ is right function and $h(y)$ is left function on interval $[c,d]$

Compound regions with intersecting curves

• Identify all curves in region and their intersection points
• Divide region into smaller subregions each bounded by pair of curves
  • Ensure each subregion has clear upper and lower (or right and left) function
• Calculate area of each subregion using appropriate integration method (with respect to x or y)
  • Use integration with respect to x for subregions bounded by $y=f(x)$ and $y=g(x)$
  • Use integration with respect to y for subregions bounded by $x=h(y)$ and $x=k(y)$
• Sum areas of all subregions to find total area of compound region

Variable selection for area integration

• Consider equations of curves when choosing integration variable
  • Integrate with respect to x for curves given as $y=f(x)$ and $y=g(x)$
  • Integrate with respect to y for curves given as $x=h(y)$ and $x=k(y)$
• Assess complexity of equations and choose variable that results in simpler expressions after solving for intersection points or integrating
• Evaluate ease of determining limits of integration and consider using variable for which limits are more easily found

Additional Considerations

• Ensure resulting expression is always non-negative when integrating with respect to x or y
  • Split integral into separate parts if necessary to maintain non-negative values ($\int_{a}^{c} [f(x)-g(x)] dx + \int_{c}^{b} [g(x)-f(x)] dx$)
• Exercise caution with curves that intersect at more than two points
  • Carefully determine appropriate intervals for integration based on desired region
• Sketch region if needed to clarify problem and identify appropriate integration method
  • Visually identify upper and lower (or right and left) functions
  • Determine points of intersection and intervals for integration

Function Analysis and Coordinate Systems

• Use the vertical line test to determine if a curve represents a function of x
• Apply the horizontal line test to check if a function is one-to-one
• Areas between curves are typically calculated in Cartesian coordinates
• Ensure the region being integrated is a bounded region to obtain a finite area