Find area bounded by `y = f^(-1)(x), x = 10, x = 4` and x-axis.
Given that area bounded by `y = f(x) , x = 2, x = 6` and x -axis is `30` sq. units, where `f(2) = 4` and `f(6) = 10`. (given `f(x)` is an invertiable function).

Answer

To find the area bounded by the curve \( y = f^{-1}(x) \), the lines \( x = 10 \), \( x = 4 \), and the x-axis, we can use the relationship between a function and its inverse. We know that the area under the curve \( y = f^{-1}(x) \) from \( x = a \) to \( x = b \) can be expressed in terms of the area under the curve \( y = f(x) \) from \( y = f^{-1}(a) \) to \( y = f^{-1}(b) \).

### Step-by-step Solution:

1. **Identify the Area to be Calculated**:
We need to calculate the area \( A \) bounded by \( y = f^{-1}(x) \), \( x = 10 \), \( x = 4 \), and the x-axis:
\[ A = \int_{4}^{10} f^{-1}(x) \, dx \]

2. **Use the Relationship Between a Function and Its Inverse**:
The area under \( f^{-1}(x) \) can be expressed as:
\[ A = \int_{4}^{10} f^{-1}(x) \, dx = \int_{f^{-1}(4)}^{f^{-1}(10)} y \, f'(y) \, dy \]
where \( f'(y) \) is the derivative of \( f(y) \).

3. **Determine the Limits**:
Given \( f(2) = 4 \) and \( f(6) = 10 \), we find:
\[ f^{-1}(4) = 2 \quad \text{and} \quad f^{-1}(10) = 6 \]

4. **Set Up the Integral**:
Substitute the limits into the integral:
\[ A = \int_{2}^{6} y \, f'(y) \, dy \]

5. **Integration by Parts**:
We can use integration by parts where:
- Let \( u = y \) and \( dv = f'(y) \, dy \)
- Then \( du = dy \) and \( v = f(y) \)

Applying integration by parts:
\[ A = \left[ y f(y) \right]_{2}^{6} - \int_{2}^{6} f(y) \, dy \]

6. **Evaluate the First Term**:
Calculate \( \left[ y f(y) \right]_{2}^{6} \):
\[ = 6 f(6) - 2 f(2) = 6 \cdot 10 - 2 \cdot 4 = 60 - 8 = 52 \]

7. **Use the Given Area**:
We know from the problem statement that:
\[ \int_{2}^{6} f(y) \, dy = 30 \]

8. **Combine the Results**:
Substitute back into the equation for \( A \):
\[ A = 52 - 30 = 22 \]

### Final Answer:
The area bounded by \( y = f^{-1}(x) \), \( x = 10 \), \( x = 4 \), and the x-axis is \( 22 \) square units.