If `y=x^2`, then area of curve y v/s x from `x=0` to 2 will be:

Options

A. `(1)/(3)`
B. `(8)/(3)`
C. `(4)/(3)`
D. `(2)/(3)`

Explanation

To find the area under the curve \( y = x^2 \) from \( x = 0 \) to \( x = 2 \), we can follow these steps:

### Step 1: Understand the area under the curve
The area under a curve \( y = f(x) \) from \( x = a \) to \( x = b \) can be calculated using the definite integral:
\[ A = \int_{a}^{b} f(x) \, dx \]

### Step 2: Set up the integral
In this case, \( f(x) = x^2 \), \( a = 0 \), and \( b = 2 \). Therefore, we set up the integral as:
\[ A = \int_{0}^{2} x^2 \, dx \]

### Step 3: Calculate the integral
To compute the integral, we use the power rule for integration:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
For our case, \( n = 2 \):
\[ \int x^2 \, dx = \frac{x^{3}}{3} + C \]
Now, we will evaluate the definite integral:
\[ A = \left[ \frac{x^3}{3} \right]_{0}^{2} \]

### Step 4: Evaluate the limits
Now we substitute the limits into the integral:
\[ A = \left( \frac{2^3}{3} \right) - \left( \frac{0^3}{3} \right) \]
Calculating this gives:
\[ A = \frac{8}{3} - 0 = \frac{8}{3} \]

### Step 5: Conclusion
Thus, the area under the curve \( y = x^2 \) from \( x = 0 \) to \( x = 2 \) is:
\[ \boxed{\frac{8}{3}} \]

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