Given fuction,`{(x^(2), "for" 0 le x lt1),(sqrtx, "for" 1le x le 2):}`
Evaluate `int_(0)^(2) f(x) d x `.

Answer

To evaluate the integral \( \int_{0}^{2} f(x) \, dx \) for the given piecewise function

\[ f(x) = \begin{cases} x^2 & \text{for } 0 \leq x < 1 \\ \sqrt{x} & \text{for } 1 \leq x \leq 2 \end{cases} \]

we will break the integral into two parts based on the definition of \( f(x) \).

### Step 1: Break the integral into two parts
We can express the integral from 0 to 2 as the sum of two integrals:

\[ \int_{0}^{2} f(x) \, dx = \int_{0}^{1} f(x) \, dx + \int_{1}^{2} f(x) \, dx \]

### Step 2: Evaluate the first integral
For \( 0 \leq x < 1 \), we have \( f(x) = x^2 \). Therefore,

\[ \int_{0}^{1} f(x) \, dx = \int_{0}^{1} x^2 \, dx \]

Using the power rule for integration, we find:

\[ \int x^2 \, dx = \frac{x^{3}}{3} + C \]

Now, we evaluate this from 0 to 1:

\[ \int_{0}^{1} x^2 \, dx = \left[ \frac{x^{3}}{3} \right]_{0}^{1} = \frac{1^{3}}{3} - \frac{0^{3}}{3} = \frac{1}{3} \]

### Step 3: Evaluate the second integral
For \( 1 \leq x \leq 2 \), we have \( f(x) = \sqrt{x} \). Therefore,

\[ \int_{1}^{2} f(x) \, dx = \int_{1}^{2} \sqrt{x} \, dx \]

We know that \( \sqrt{x} = x^{1/2} \), so we can integrate:

\[ \int \sqrt{x} \, dx = \int x^{1/2} \, dx = \frac{x^{3/2}}{3/2} + C = \frac{2}{3} x^{3/2} + C \]

Now, we evaluate this from 1 to 2:

\[ \int_{1}^{2} \sqrt{x} \, dx = \left[ \frac{2}{3} x^{3/2} \right]_{1}^{2} = \frac{2}{3} (2^{3/2}) - \frac{2}{3} (1^{3/2}) = \frac{2}{3} (2\sqrt{2}) - \frac{2}{3} (1) \]

Calculating further:

\[ = \frac{2 \cdot 2\sqrt{2}}{3} - \frac{2}{3} = \frac{4\sqrt{2}}{3} - \frac{2}{3} \]

### Step 4: Combine the results
Now, we combine the results of both integrals:

\[ \int_{0}^{2} f(x) \, dx = \int_{0}^{1} f(x) \, dx + \int_{1}^{2} f(x) \, dx = \frac{1}{3} + \left( \frac{4\sqrt{2}}{3} - \frac{2}{3} \right) \]

This simplifies to:

\[ = \frac{1}{3} + \frac{4\sqrt{2}}{3} - \frac{2}{3} = \frac{4\sqrt{2} - 1}{3} \]

Thus, the final answer is:

\[ \int_{0}^{2} f(x) \, dx = \frac{4\sqrt{2} - 1}{3} \]