Definite integrals and signed area
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Area Under a Curve
Interpret definite integrals as signed area and calculate total area carefully.
Signed area
A definite integral gives signed area: area above the \(x\)-axis counts as positive and area below the \(x\)-axis counts as negative.
\[ \int_a^b f(x)\,dx=\text{signed area between the curve and the }x\text{-axis} \]
Worked example
Find the total area between \(y=x^2-4\) and the \(x\)-axis from \(x=1\) to \(x=3\).
- The curve crosses the \(x\)-axis at \(x=2\).
- Total area is \(-\int_1^2(x^2-4)\,dx+\int_2^3(x^2-4)\,dx\).
- Using \(F(x)=\frac{x^3}{3}-4x\), this gives \(\frac{5}{3}+\frac{7}{3}=4\).
Area under a curve
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Positive and negative area
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