Suppose $f(x)$ is a piecewise function defined as $f(x)=7 g(x)$ for $x<4$ $f(x)=8$ for $x \geq 4$ If $\int_{0}^{4} g(x) d x=5$ then $\int_{0}^{21} f(x) d x=$

Step 1 ：Define the function \(f(x)\) as a piecewise function: \(f(x)=7 g(x)\) for \(x<4\) and \(f(x)=8\) for \(x \geq 4\)

Step 2 ：Given that \(\int_{0}^{4} g(x) d x=5\)

Step 3 ：Calculate the integral of \(f(x)\) from 0 to 4: \(\int_{0}^{4} f(x) dx = 7 \times \int_{0}^{4} g(x) dx = 7 \times 5 = 35\)

Step 4 ：Calculate the integral of \(f(x)\) from 4 to 21: \(\int_{4}^{21} f(x) dx = 8 \times (21-4) = 136\)

Step 5 ：Add the two parts to get the integral of \(f(x)\) from 0 to 21: \(\int_{0}^{21} f(x) dx = \int_{0}^{4} f(x) dx + \int_{4}^{21} f(x) dx = 35 + 136 = 171\)

Step 6 ：Final Answer: \(\boxed{171}\)