https://file.aiquickdraw.com/mathbot/file/1729036441952gcl6wvq9.jpg

Let's analyze the problem in the image. 

---

The problem gives:
\[
f(x) = \sqrt{x}, \quad \text{continua en el intervalo} \; [1,2],
\]
and states that:
\[
F(x) = \frac{2}{3} \sqrt{x^3}.
\]
We are asked to evaluate the definite integral:
\[
\int_1^2 \sqrt{x} \, dx.
\]

---

### Step 1: Integrate \( f(x) = \sqrt{x} = x^{1/2} \).
We apply the power rule of integration:
\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad \text{for} \; n \neq -1.
\]
Here, \( n = \frac{1}{2} \), so:
\[
\int \sqrt{x} \, dx = \int x^{1/2} \, dx = \frac{x^{3/2}}{3/2} + C = \frac{2}{3} x^{3/2} + C.
\]

---

### Step 2: Apply the Fundamental Theorem of Calculus.
We need to evaluate the definite integral:
\[
\int_1^2 \sqrt{x} \, dx = \left[ \frac{2}{3} x^{3/2} \right]_1^2.
\]

Now, substitute the limits \( 1 \) and \( 2 \) into the antiderivative.

1. **Upper limit** \( x = 2 \):
   \[
   \frac{2}{3} \cdot 2^{3/2} = \frac{2}{3} \cdot \sqrt{8} = \frac{2}{3} \cdot 2\sqrt{2} = \frac{4\sqrt{2}}{3}.
   \]

2. **Lower limit** \( x = 1 \):
   \[
   \frac{2}{3} \cdot 1^{3/2} = \frac{2}{3} \cdot 1 = \frac{2}{3}.
   \]

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### Step 3: Compute the difference.
\[
\int_1^2 \sqrt{x} \, dx = \frac{4\sqrt{2}}{3} - \frac{2}{3}.
\]
Factor out \( \frac{2}{3} \):
\[
\frac{2}{3} \left( 2\sqrt{2} - 1 \right).
\]

---

### Final Answer:
\[
\int_1^2 \sqrt{x} \, dx = \frac{2}{3} \left( 2\sqrt{2} - 1 \right).
\]

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Do you have any questions, or would you like further clarification on a specific part?  

### 5 Related Questions:
1. How do you apply the Fundamental Theorem of Calculus in general cases?
2. What is the power rule for integration and when does it apply?
3. How do you simplify radicals like \( \sqrt{8} \) to \( 2\sqrt{2} \)?
4. How do definite integrals differ from indefinite integrals?
5. What happens if the limits of integration are reversed?

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**Tip:** Always simplify radicals and fractions carefully to avoid computational errors when evaluating definite integrals.

Evaluate the definite integral ∫_1^2 √x dx where f(x) = √x, continuous on the interval [1, 2], and F(x) = (2/3) √x^3.

Learn how to solve the definite integral ∫_1^2 √x dx step-by-step. This problem involves applying the power rule for integration and using the Fundamental Theorem of Calculus to compute the result. Suitable for college-level calculus students.

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