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To find the shaded area under the curve \( f(x) = \sqrt{x + 3} \), we’ll set up an integral over the specified region. 

### Step-by-Step Solution

1. **Determine the Limits of Integration**:
   - From the graph, we see that the shaded region starts at \( x = -3 \) (where \( f(-3) = 0 \)) and ends at \( x = 1 \) (where \( f(1) = \sqrt{4} = 2 \)).
   
2. **Set Up the Integral**:
   - The area under the curve from \( x = -3 \) to \( x = 1 \) can be calculated using:
     \[
     \text{Area} = \int_{-3}^{1} f(x) \, dx = \int_{-3}^{1} \sqrt{x + 3} \, dx
     \]

3. **Evaluate the Integral**:
   - To integrate \( \sqrt{x + 3} \), we can use the substitution \( u = x + 3 \), so \( du = dx \) and the limits change as follows:
     - When \( x = -3 \), \( u = 0 \)
     - When \( x = 1 \), \( u = 4 \)
   
   - The integral becomes:
     \[
     \int_{-3}^{1} \sqrt{x + 3} \, dx = \int_{0}^{4} \sqrt{u} \, du
     \]

4. **Integrate \( \sqrt{u} \)**:
   - We know that \( \sqrt{u} = u^{1/2} \), and the integral of \( u^{1/2} \) is:
     \[
     \int u^{1/2} \, du = \frac{2}{3} u^{3/2}
     \]
   - Applying this result from \( u = 0 \) to \( u = 4 \):
     \[
     \int_{0}^{4} \sqrt{u} \, du = \left[ \frac{2}{3} u^{3/2} \right]_{0}^{4}
     \]
   - Substitute the limits:
     \[
     = \frac{2}{3} (4)^{3/2} - \frac{2}{3} (0)^{3/2}
     \]
     \[
     = \frac{2}{3} \cdot 8 - 0 = \frac{16}{3}
     \]

### Final Answer
The shaded area is:
\[
\text{Area} = \frac{16}{3}
\]

Would you like a deeper explanation of any step?

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Here are some related questions to deepen understanding:

1. How do you determine the limits of integration for an area under a curve?
2. Why is substitution useful when integrating functions with square roots?
3. What is the geometric interpretation of integrating a function from one point to another?
4. How would the approach change if the shaded area extended beyond the origin?
5. Can you explain why \( \frac{2}{3} u^{3/2} \) is the antiderivative of \( u^{1/2} \)?

**Tip**: When solving definite integrals, carefully substitute limits after applying any transformations or substitutions.

Find the shaded area under the curve f(x) = √(x + 3) from x = -3 to x = 1.

Calculate the shaded area under the curve f(x) = √(x + 3) between x = -3 and x = 1 using definite integrals. This solution covers key calculus concepts, including integration techniques and area determination under a curve, suitable for Grades 11-12.

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