Calculate the area under the curve `y=2sqrtx` included between the lines x = 0 and x = 1.

Explanation

To calculate the area under the curve \( y = 2\sqrt{x} \) between the lines \( x = 0 \) and \( x = 1 \), we will follow these steps:

### Step 1: Set up the integral
The area \( A \) under the curve from \( x = 0 \) to \( x = 1 \) can be expressed as:
\[ A = \int_{0}^{1} (2\sqrt{x}) \, dx \]

### Step 2: Simplify the integrand
We can rewrite \( 2\sqrt{x} \) as \( 2x^{1/2} \):
\[ A = \int_{0}^{1} 2x^{1/2} \, dx \]

### Step 3: Integrate the function
Using the power rule for integration, we have:
\[ \int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C \]
For our case, \( n = \frac{1}{2} \):
\[ \int 2x^{1/2} \, dx = 2 \cdot \frac{x^{3/2}}{3/2} = \frac{4}{3} x^{3/2} \]

### Step 4: Evaluate the definite integral
Now we evaluate this from \( 0 \) to \( 1 \):
\[ A = \left[ \frac{4}{3} x^{3/2} \right]_{0}^{1} = \frac{4}{3} (1^{3/2}) - \frac{4}{3} (0^{3/2}) \]
Calculating this gives:
\[ A = \frac{4}{3} (1) - \frac{4}{3} (0) = \frac{4}{3} - 0 = \frac{4}{3} \]

### Final Answer
Thus, the area under the curve \( y = 2\sqrt{x} \) from \( x = 0 \) to \( x = 1 \) is:
\[ \boxed{\frac{4}{3}} \text{ square units} \]