8 - Calculus

Question

Shade in the regions between the two functions shown here on the intervals (a) $[-2,3]$; (b) $[-1,2]$and (c)$[1,3]$. Which of these regions has the largest area? The smallest?

Step-by-Step Solution

Answer

Since ${\int }_{-3}^{2}f\left(x\right)dx$ is negative obviously $-{\int }_{-3}^{2}f\left(x\right)dx$ will be positive. Therefore, the answers are negative, positive.

Step 1: Given information

the regions between the two functions shown here on the intervals (a) $[-2,3]$ ; (b) [-1,2] and (c) $[1,3]$

Step 2: Calculation

It is supposed that $f$ is negative on $[-3,2]$.

The objective is to determine if ${\int }_{-3}^{2}f\left(x\right)dx$ and $-{\int }_{-3}^{2}f\left(x\right)dx$ will be positive or negative. Since, $f$is negative on $[-3,2]$ so ${\int }_{-3}^{2}f\left(x\right)dx$ will also be negative.

Now, since ${\int }_{-3}^{2}f\left(x\right)dx$ is negative obviously $-{\int }_{-3}^{2}f\left(x\right)dx$ will be positive. Therefore, the answers are negative, positive.