Suppose that $f$ is a continuous function such that $f(x) \geq 0$ for $a \leq x \leq b$, and suppose that the area under the curve $y=f(x)$ between $x=a$ and $x=b$ is 4 . If $F$ is an antiderivative of $f$ for which $F(a)=7$, then what is the value of $F(b)$ ?

Step 1 ：Given that $F$ is an antiderivative of $f$ and $F(a)=7$.

Step 2 ：We also know that the area under the curve $y=f(x)$ between $x=a$ and $x=b$ is 4.

Step 3 ：According to the Fundamental Theorem of Calculics, if a function $f$ is continuous over the interval $[a, b]$ and $F$ is an antiderivative of $f$ on $[a, b]$, then $\int_{a}^{b} f(x) dx = F(b) - F(a)$.

Step 4 ：Substituting the given values into the equation, we get $4 = F(b) - 7$.

Step 5 ：Solving for $F(b)$, we find that $F(b) = 11$.

Step 6 ：Final Answer: The value of $F(b)$ is \(\boxed{11}\).