Areas Under Curves

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The area bounded by y = x² and y = x from x = 0 to x = 1 is

RankGuru · Accepted Answer

1. **Identify Upper and Lower Curves**: On the interval $(0, 1)$, $x \ge x^2$. So $y=x$ is the upper curve and $y=x^2$ is the lower curve.
2. **Setup Area Integral**: $Area = \int_0^1 (Upper - Lower) \, dx = \int_0^1 (x - x^2) \, dx$.
3. **Antiderivative**: $[\frac{x^2}{2} - \frac{x^3}{3}]_0^1$
4. **Evaluate**:
   - $(\frac{1}{2} - \frac{1}{3}) - (0 - 0)$
   - $= \frac{3-2}{6} = 1/6$.
5. **Result**: 1/6 square units.

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The area under the curve y = x from x = 0 to x = 2 is

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