Question

Suppose f is positive on (−∞, −1] and [2,∞) and negative on the interval [−1, 2]. Write (a) the signed area and (b) the absolute area between the graph of f and the x-axis on [−3, 4] in terms of definite integrals that do not involve absolute values.

Short answer

(a). ${\int }_{-3}^{4}f\left(x\right)dx$

(b).${\int }_{-3}^{-1} f\left(x\right)dx-{\int }_{-1}^2 f\left(x\right)dx+{\int }_2^4 f\left(x\right)dx$

Step-by-step solution

(a). Given Information

$\text{A function}f\text{is positive on}(-\mathrm{\infty },-1];[2,\mathrm{\infty })\text{and negative on}[-1,2]$

$\text{The objective is to write the signed area on}[-3,4]\text{.}$

$$\begin{gathered} \text{The signed area will be,} \\ {\int }_{-3}^{4}f\left(x\right)dx \end{gathered}$$

$\text{Therefore, the signed area is}{\int }_{-3}^4 f\left(x\right)dx\text{.}$

(b). Given Information

$\text{The objective is to write the absolute area on}[-3,4]\text{without using the absolute area absolute}$ values.

$\text{The intervals will be}[-3,-1],[-1,2],[2,4]$

$\text{The absolute area will be}$

${\int }_{-3}^{-1} f\left(x\right)dx-{\int }_{-1}^2 f\left(x\right)dx+{\int }_2^4 f\left(x\right)dx$

$\text{Therefore, the signed area is}{\int }_{-3}^{-1} f\left(x\right)dx-{\int }_{-1}^2 f\left(x\right)dx+{\int }_2^4 f\left(x\right)dx\text{.}$