QuestionSuppose that $F^{\prime}(x)=f(x)$ and $F(0)=3, F(2)=7$. a. What is the area under $y=f(x)$ over $[0,2]$ if $f(x) \geq 0$ ? b. What is the graphical interpretation of $F(2)-F(0)$ if $f(x)$ takes on both positive and negative values?

Studdy Solution
The graphical interpretation of $F(2) - F(0)$ when $f(x)$ takes on both positive and negative values is as follows:
- $F(2) - F(0)$ represents the net area between the curve $y = f(x)$ and the x-axis over the interval $[0,2]$. - The net area is the total area above the x-axis minus the total area below the x-axis. - If $f(x)$ is positive, it contributes positively to the net area; if $f(x)$ is negative, it contributes negatively.
Thus, $F(2) - F(0)$ gives the net signed area under the curve $y = f(x)$ from $x = 0$ to $x = 2$.
The area under $y = f(x)$ over $[0,2]$ is $\boxed{4}$, and $F(2) - F(0)$ represents the net signed area under the curve when $f(x)$ has both positive and negative values.