Areas Between Two Curves
Areas Between Two Curves
Find all intersections bounding the required region. A vertical strip has height equal to the upper curve minus the lower curve, even when both lie below the horizontal axis.
\(A=\int_a^b[\mathrm{f}(x)-\mathrm{g}(x)]\,\mathrm{d}x\), provided \(\mathrm{f}(x)\geq\mathrm{g}(x)\) throughout the interval.
For \(y=2x\) and \(y=x^2\), intersections satisfy \(x(x-2)=0\). On \(0\leq x\leq2\), the line is above the parabola.
\(A=\int_0^2(2x-x^2)\,\mathrm{d}x=\left[x^2-\dfrac{x^3}{3}\right]_0^2=\dfrac43\).
If the curves exchange order inside the interval, split there and reverse the subtraction on the next piece. State area in square units.
For horizontal strips, write the boundaries as \(x=R(y)\) and \(x=L(y)\), then integrate \(A=\int_c^d[R(y)-L(y)]\,\mathrm{d}y\), with right minus left and \(y\)-limits. Split where their order changes. This horizontal-strip method is optional enrichment in AA SL; vertical strips are the SL core method.
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