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8.4 Finding the Area Between Curves Expressed as Functions of x

Part of Applications of Integration in AP® Calculus BC. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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When two curves are both written as functions of x, the area between them on an interval is found by integrating top minus bottom with respect to x. The formula is
∫ab[f(x)−g(x)] dx\int_a^b [f(x)-g(x)]\,dx
where f(x) is the upper curve and g(x) is the lower curve on [a,b]. This works because each vertical slice has height equal to the difference between the two functions and width dx, so the integral adds up all those slice areas.

The most important step is identifying which curve is on top over the interval. If you subtract in the wrong order, you may get a negative value even though area must be positive. You should also find the intersection points when they are not given, since they often determine the limits of integration. This is one of the most common geometry applications of definite integrals, and it extends the idea of area under a single curve to area trapped between two graphs.

In this unit

13 lessons

  1. 8.1Finding the Average Value of a Function on an IntervalPreview lesson
  2. 8.2Connecting Position, Velocity, and Acceleration of Functions Using IntegralsPreview lesson
  3. 8.3Using Accumulation Functions and Definite Integrals in Applied ContextsPreview lesson
  4. 8.4Finding the Area Between Curves Expressed as Functions of xThis lesson
  5. 8.5Finding the Area Between Curves Expressed as Functions of yPreview lesson
  6. 8.6Finding the Area Between Curves That Intersect at More Than Two PointsPreview lesson
  7. 8.7Volumes with Cross Sections: Squares and RectanglesPreview lesson
  8. 8.8Volumes with Cross Sections: Triangles and SemicirclesPreview lesson
  9. 8.9Volume with Disc Method: Revolving Around the x- or y-AxisPreview lesson
  10. 8.10Volume with Disc Method: Revolving Around Other AxesPreview lesson
  11. 8.11Volume with Washer Method: Revolving Around the x- or y-AxisPreview lesson
  12. 8.12Volume with Washer Method: Revolving Around Other AxesPreview lesson
  13. 8.13The Arc Length of a Smooth, Planar Curve and Distance TraveledPreview lesson

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