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Calculus learning

Area Between Curves: Intersections, Top Minus Bottom and Splitting

Area between curves is not always the signed integral of their difference. First locate intersections inside the interval, then decide which graph is higher on each segment. Add the segment areas without cancelling regions on opposite sides.

Direct answer

A=∫ₐᵇ |f(x)−g(x)| dx. Split wherever the upper and lower curves exchange places.

The curves must be integrable on the chosen finite interval. Geometric area is nonnegative and uses square units.

Area Between Curves worked method

For y=x and y=x² on [0,1], solve x=x² to get intersections 0 and 1. In between, x≥x².

Integrate x−x²: [x²/2−x³/3]₀¹=1/6. The result is in square units, not units of slope.

For y=x and y=0 on [−1,1], the curves cross at 0. Integrate −x on [−1,0] and x on [0,1]; each contributes 1/2.

The signed integral ∫₋₁¹ x dx is zero, while the geometric area is one. Taking |0| afterward does not restore the two cancelled pieces.

If solving for y gives easier boundaries, use horizontal slices and integrate right-minus-left in dy. Do not mix x-bounds and y-bounds.

1
A=∫₀¹ (x−x²) dx=1/6
2
A=∫₋₁⁰ (−x) dx+∫₀¹ x dx=1
3
Signed integral ≠ total geometric area

Worked examples

Between x and x², 0≤x≤1
16\frac{1}{6}

Find intersections before selecting the higher curve.

Between x and 0, −1≤x≤1
11

Split at the crossing.

Between 2 and x², 0≤x≤1
53\frac{5}{3}

The constant curve stays above the parabola.

Common mistakes

  • • Taking the absolute value only after cancellation.
  • • Using only outer endpoints when curves cross inside.
  • • Reporting square units as cubic units.

Try the supported calculator preset

The preset opens a supported expression with its variable and conditions. Teaching examples elsewhere in this guide are reference explanations, not claims that the engine supports every method.

The ordinary definite-integral preset is valid because x−x² is nonnegative on [0,1]. The application tool handles supported crossings automatically.

Open the area and volume application toolCalculadora de integral definida:x−x2x - x^{2}
Open the calculator without a preset: Calculadora de integral definida

References

Frequently asked questions

Can area be negative?+

No. A negative integral signals orientation or a signed quantity, not geometric area.

Do I always need to split?+

Only where the ordering changes; intersections that merely touch need not change the sign.

What does the application tool accept?+

Finite quadratic-or-lower polynomials. It finds their real crossings rather than guessing signs from a few samples.

Conclusion

Area Between Curves starts with the stated domain and conditions. Recheck those before carrying a worked example into a different problem.

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