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AP Calculus AB · Unit 8: Applications of Integration

8.6 Area with Multiple Intersections

Split the region

When two curves intersect more than twice, split the region into subintervals at each intersection. Between intersections, the top and bottom curves can switch.

Sum the pieces

On each subinterval, integrate (top − bottom) for that piece, then add the areas. Identify which curve is on top separately on each piece.

Key terms

Multiple intersections
Split the region at each crossing.
Top/bottom switch
Which curve is higher can change between crossings.
Each piece
Integrate (top − bottom) for that subinterval.
Total
Sum the areas of all pieces.
Identify top
Determine the upper curve on each subinterval.

Try an AP-style question

What is the total area between y = x³ and y = x?

  1. 1/2
  2. 1
  3. 0
  4. 1/4
Show the answer

A. They meet at −1, 0, 1. By symmetry: 2∫₀¹ (x − x³) dx = 2(1/4).

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