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8.6 Finding the Area Between Curves That Intersect at More Than Two Points

Syllabus
2020
Topic
8.6
Level
—

Learning objectives

—CHA-5.A—Calculate areas in the plane using the definite integral—Topic 8.6• CHA-5.A Calculate areas in the plane using the definite integral.• CHA-5.A.3 Areas of certain regions in the plane may be calculated using a sum of two or more definite integrals or by evaluating a definite integral of the absolute value of the difference of two functions.• Enduring understanding CHA-5: Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.

Split Area Wherever the Curves Exchange Order

If two curves intersect several times, their upper–lower order may change at each intersection. Use every relevant intersection as a possible partition point, test the sign of f−gf-g on each subinterval, and make each slice height nonnegative.

A=\sum_i\int_{x_i}^{x_{i+1}}(\text{upper}-\text{lower}),dx=\int_a^b|f(x)-g(x)|,dx

  1. Solve f(x)=g(x)f(x)=g(x) for all intersections in the interval.
  2. Order those xx-values.
  3. Test one point in each subinterval.
  4. Write upper minus lower on each piece, or retain one absolute-difference integral.
  5. Add all nonnegative contributions.

On [0,2π][0,2\pi], y=sin⁡xy=\sin x and y=0y=0 intersect at 00, π\pi, and 2π2\pi. Since sin⁡x≥0\sin x\ge0 on [0,π][0,\pi] and sin⁡x≤0\sin x\le0 on [π,2π][\pi,2\pi], A=∫0πsin⁡x dx+∫π2π(−sin⁡x) dx=2+2=4A=\int_0^\pi\sin x\,dx+\int_\pi^{2\pi}(-\sin x)\,dx=2+2=4. Equivalently, A=∫02π∣sin⁡x∣ dx=4A=\int_0^{2\pi}|\sin x|\,dx=4.

The unsplit signed integral ∫02πsin⁡x dx=0\int_0^{2\pi}\sin x\,dx=0 is net signed area, not total geometric area. Absolute value must be applied to the height before integration, not to the final signed integral after positive and negative regions have already canceled.