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Area under the curve

Area under a Curve

  • The area between the graph of \(y = f(x)\) and the \(x\)-axis (shown below) is given by the definite integral:
    \[ \mbox{Area}=\int_{a}^{b} f(x) \; \mathrm{d} x\]

  • This formula gives a positive result for a graph above the \(x\)-axis, and a negative result for a graph below the \(x\)-axis.
  • If the graph of \(y=f(x)\) is partly above and partly below the
    \(x\)-axis, the definite integral generates the overall (net) area. That is, the area above the axis minus the area below the axis. In more complicated cases it may be necessary to split the area into parts and add or subtract parts to calculate areas.

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