The area bounded by the curve `x=f(y)`, the Y-axis and the two lines `y=a` and y=b is equal to

Options

A. `int_(a)^(b)ydx`
B. `int_(a)^(b)y^(2)dx`
C. `int_(a)^(b)xdy`
D. None of the above

Answer

To find the area bounded by the curve \( x = f(y) \), the Y-axis, and the two lines \( y = a \) and \( y = b \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Boundaries**:
- The area is bounded by the curve \( x = f(y) \), the Y-axis (which is \( x = 0 \)), and the horizontal lines \( y = a \) and \( y = b \).

2. **Set Up the Integral**:
- The area \( A \) can be expressed as the integral of the function \( f(y) \) from \( y = a \) to \( y = b \). Since the area is to the right of the Y-axis, we consider the integral of \( f(y) \) with respect to \( y \).
- The formula for the area is:
\[ A = \int_{a}^{b} f(y) \, dy \]

3. **Evaluate the Integral**:
- To find the area, you would compute the integral \( \int_{a}^{b} f(y) \, dy \). This integral will give you the total area under the curve from \( y = a \) to \( y = b \).

4. **Final Result**:
- The area bounded by the curve \( x = f(y) \), the Y-axis, and the lines \( y = a \) and \( y = b \) is given by:
\[ A = \int_{a}^{b} f(y) \, dy \]