Computing areas using integrals

Area of a “curvilinear trapezoid”

Let $f$ be nonnegative and continuous on $[a;b]$ . Then the geometric area of the region under $y=f(x)$ , above $Ox$ , and between $x=a$ and $x=b$ , equals $\displaystyle\int_a^b f(x)\,dx$ . If $f$ goes below the axis on part of the segment, $\int_a^b f$ gives signed area: parts below the axis count negatively. For “plain magnitude” area between the curve and the axis use $\int_a^b |f(x)|\,dx$ or split into intervals of constant sign and add absolute values.

Always align the picture with the sign of $f$ : do not confuse $\int$ and “shaded area”

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