What is the area under the curve g(x) from *x*=21 to x=0?

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.

Learn how to calculate the total area under the curve g(x) from x = 21 to x = 0 using integral calculus. This detailed solution includes step-by-step calculations for triangular and semicircular areas. Suitable for advanced high school students and college-level learners.

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