A definite integral adds signed contributions: positive above the horizontal axis, negative below it. Geometric area counts both regions positively.
\(A=\int_a^b|\mathrm{f}(x)|\,\mathrm{d}x\). Split at every zero where the sign changes.
For \(y=x-1\) on \(0\leq x\leq3\):
\(\int_0^3(x-1)\,\mathrm{d}x=\dfrac32\), whereas \(A=-\int_0^1(x-1)\,\mathrm{d}x+\int_1^3(x-1)\,\mathrm{d}x=\dfrac52\).
Taking the absolute value after integrating gives \(\left|\int_a^b\mathrm{f}(x)\,\mathrm{d}x\right|\), which can be smaller than the total area. Check the sign on each interval before evaluating.
For a modulus integrand, replace \(|\mathrm{f}(x)|\) by \(\mathrm{f}(x)\) where it is non-negative and by \(-\mathrm{f}(x)\) where it is negative.
Need help? Join our JC Math tuition classes.
Learn more