Area bounded by the parabola `3y=x^(2)`, the X-axis and the lines x = 2, x = 3 is

Options

A. `(29)/(9)`
B. `(17)/(9)`
C. `(7)/(9)`
D. `(19)/(9)`

Answer

To find the area bounded by the parabola \(3y = x^2\), the x-axis, and the lines \(x = 2\) and \(x = 3\), we will follow these steps:

### Step 1: Rewrite the equation of the parabola
The equation of the parabola can be rewritten in terms of \(y\):
\[ y = \frac{x^2}{3} \]

### Step 2: Set up the integral for the area
The area \(A\) between the curve \(y = \frac{x^2}{3}\) and the x-axis from \(x = 2\) to \(x = 3\) can be calculated using the definite integral:
\[ A = \int_{2}^{3} \frac{x^2}{3} \, dx \]

### Step 3: Calculate the integral
First, we can factor out the constant \(\frac{1}{3}\):
\[ A = \frac{1}{3} \int_{2}^{3} x^2 \, dx \]

Now, we need to compute the integral \(\int x^2 \, dx\):
\[ \int x^2 \, dx = \frac{x^3}{3} \]

### Step 4: Evaluate the definite integral
Now we can evaluate the definite integral from \(2\) to \(3\):
\[ \int_{2}^{3} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{2}^{3} = \frac{3^3}{3} - \frac{2^3}{3} = \frac{27}{3} - \frac{8}{3} = \frac{27 - 8}{3} = \frac{19}{3} \]

### Step 5: Multiply by the constant factor
Now, substituting back into the area formula:
\[ A = \frac{1}{3} \cdot \frac{19}{3} = \frac{19}{9} \]

### Conclusion
Thus, the area bounded by the parabola \(3y = x^2\), the x-axis, and the lines \(x = 2\) and \(x = 3\) is:
\[ \boxed{\frac{19}{9}} \text{ square units} \]

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