The area bounded by the curve `y=(x^(2))/(4)+(x)/(4)-(1)/(2)` with X axis in [0, 2] is

Options

A. a) `(1)/(6)`
B. b) `(2)/(3)`
C. c) `(4)/(3)`
D. d) 3

Answer

To find the area bounded by the curve \( y = \frac{x^2}{4} + \frac{x}{4} - \frac{1}{2} \) and the x-axis in the interval [0, 2], we will follow these steps:

### Step 1: Identify the curve and the interval
The given curve is a quadratic function, and we need to find the area between this curve and the x-axis from \( x = 0 \) to \( x = 2 \).

### Step 2: Calculate the values of the function at the endpoints
We will evaluate the function at the endpoints of the interval:

1. **At \( x = 0 \)**:
\[ y(0) = \frac{0^2}{4} + \frac{0}{4} - \frac{1}{2} = -\frac{1}{2} \]

2. **At \( x = 2 \)**:
\[ y(2) = \frac{2^2}{4} + \frac{2}{4} - \frac{1}{2} = \frac{4}{4} + \frac{2}{4} - \frac{1}{2} = 1 + \frac{1}{2} - \frac{1}{2} = 1 \]

### Step 3: Set up the integral for the area
The area \( A \) bounded by the curve and the x-axis from \( x = 0 \) to \( x = 2 \) can be calculated using the definite integral:
\[ A = \int_{0}^{2} \left( \frac{x^2}{4} + \frac{x}{4} - \frac{1}{2} \right) \, dx \]

### Step 4: Break down the integral
We can separate the integral into three parts:
\[ A = \int_{0}^{2} \frac{x^2}{4} \, dx + \int_{0}^{2} \frac{x}{4} \, dx - \int_{0}^{2} \frac{1}{2} \, dx \]

### Step 5: Calculate each integral
1. **First integral**:
\[ \int_{0}^{2} \frac{x^2}{4} \, dx = \frac{1}{4} \cdot \left[ \frac{x^3}{3} \right]_{0}^{2} = \frac{1}{4} \cdot \left( \frac{2^3}{3} - 0 \right) = \frac{1}{4} \cdot \frac{8}{3} = \frac{2}{3} \]

2. **Second integral**:
\[ \int_{0}^{2} \frac{x}{4} \, dx = \frac{1}{4} \cdot \left[ \frac{x^2}{2} \right]_{0}^{2} = \frac{1}{4} \cdot \left( \frac{2^2}{2} - 0 \right) = \frac{1}{4} \cdot 2 = \frac{1}{2} \]

3. **Third integral**:
\[ \int_{0}^{2} \frac{1}{2} \, dx = \frac{1}{2} \cdot [x]_{0}^{2} = \frac{1}{2} \cdot (2 - 0) = 1 \]

### Step 6: Combine the results
Now we can combine the results of the integrals:
\[ A = \frac{2}{3} + \frac{1}{2} - 1 \]

### Step 7: Simplify the expression
To simplify \( A \):
1. Convert \( \frac{1}{2} \) and \( 1 \) to have a common denominator of 6:
\[ A = \frac{2}{3} + \frac{3}{6} - \frac{6}{6} = \frac{2}{3} + \frac{3 - 6}{6} = \frac{2}{3} - \frac{3}{6} \]
2. Convert \( \frac{2}{3} \) to sixths:
\[ \frac{2}{3} = \frac{4}{6} \]
3. Combine:
\[ A = \frac{4}{6} - \frac{3}{6} = \frac{1}{6} \]

### Final Answer
The area bounded by the curve and the x-axis from \( x = 0 \) to \( x = 2 \) is \( \frac{1}{6} \).

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