5.1 · Area between curves and average value

A slice measures a difference in heights

For vertical slices, area is $\displaystyle \int_a^b(\mathrm{top}-\mathrm{bottom})\,dx$. First find curve intersections and inspect which graph is above on each interval. Geometric area cannot be negative.

For horizontal slices, use right minus left and integrate with respect to $\displaystyle y$. Choose the direction that avoids unnecessary pieces, and express all slice dimensions in the chosen variable.

$$A=\int_a^b(f(x)-g(x))\,dx\quad\mathrm{when}\quad f\geq g$$

Split when the ordering changes

The area between $\displaystyle y=x$ and the axis on $\displaystyle [-1,1]$ is not its signed integral, which is zero. Split at the crossing and reverse the difference on the negative side, or integrate the absolute value.

Always solve the intersection equation in the stated domain. A sketch helps identify extra intersections that a convenient single interval would miss.

Average value is accumulation divided by width

The average value of a continuous $\displaystyle f$ on $\displaystyle [a,b]$ is $\displaystyle (1/(b-a))\int_a^bf(x)\,dx$. Its units match the function values, since dividing by interval width removes the input unit.

The average of the two endpoint values generally differs from the average function value. Equality holds for a linear function, but not for an arbitrary curve.

$$f_{\mathrm{avg}}=\frac{1}{b-a}\int_a^b f(x)\,dx$$