To find the total area under the curve $g(x)$ from $x = 0$ to $x = 21$, we need to sum up the absolute values of the areas of the individual sections of the curve:

1. From $x = 0$ to $x = 6$:

   • This section forms a right triangle with base 6 and height 12.
   • Area: $\frac{1}{2} \times 6 \times 12 = 36$

2. From $x = 6$ to $x = 18$:

   • This section forms a semicircle with radius 6.
   • Area of the semicircle: $\frac{1}{2} \pi (6)^2 = 18\pi$

3. From $x = 18$ to $x = 21$:

   • This section forms a right triangle with base 3 and height 6.
   • Area: $\frac{1}{2} \times 3 \times 6 = 9$

Calculating Total Area:

1. Area from $x = 0$ to $x = 6$: $A_1 = 36$

2. Area from $x = 6$ to $x = 18$: $A_2 = 18\pi$

3. Area from $x = 18$ to $x = 21$: $A_3 = 9$

Total Area Under the Curve:

$A_{\text{total}} = A_1 + |A_2| + A_3 = 36 + 18\pi + 9$

Therefore, the total area under the curve $g(x)$ from $x = 0$ to $x = 21$ is: $A_{\text{total}} = 45 + 18\pi$

Would you like any details or have any questions on this solution?

Here are 8 related questions for further practice:

1. What is the definite integral of $g(x)$ from $x = 0$ to $x = 21$?
2. How does the sign of the area change when integrating below the x-axis?
3. Can you calculate the area under the curve for a different piecewise function?
4. How do you interpret the area under the curve for a function that changes sign within the interval?
5. What methods can be used to approximate the area under a curve if the function is not easily integrable?
6. How would you set up the integral for a function with both linear and non-linear segments?
7. What is the importance of understanding areas under curves in real-world applications?
8. How do you handle integrals involving piecewise continuous functions in multiple intervals?

Tip: When calculating the total area under a curve, be careful to treat each section separately and take the absolute value of areas below the x-axis to ensure a correct total area.