Q. 5 - Calculus

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. 

Step-by-Step Solution

Answer

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

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

(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.

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

$\text{ The absolute area will be }$

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

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