If `int_(0)^(1)[4x^(3)-f(x)dx=(4)/(7)]` then find the area of region bounded by `y=f(x)`, x-axis and coordinate x = 1 and x = 2.

Answer

To solve the problem step by step, we will follow the instructions provided in the video transcript.

### Step 1: Evaluate the given integral
We start with the equation given in the problem:

\[ \int_{0}^{1} (4x^3 - f(x)) \, dx = \frac{4}{7} \]

We can separate this into two integrals:

\[ \int_{0}^{1} 4x^3 \, dx - \int_{0}^{1} f(x) \, dx = \frac{4}{7} \]

### Step 2: Calculate the integral of \(4x^3\)
Now, we calculate the integral of \(4x^3\):

\[ \int 4x^3 \, dx = 4 \cdot \frac{x^4}{4} = x^4 \]

Now we evaluate this from 0 to 1:

\[ \left[ x^4 \right]_{0}^{1} = 1^4 - 0^4 = 1 - 0 = 1 \]

### Step 3: Substitute back into the equation
Substituting this back into our equation gives:

\[ 1 - \int_{0}^{1} f(x) \, dx = \frac{4}{7} \]

### Step 4: Solve for \(\int_{0}^{1} f(x) \, dx\)
Rearranging this equation, we find:

\[ \int_{0}^{1} f(x) \, dx = 1 - \frac{4}{7} = \frac{3}{7} \]

### Step 5: Find the area bounded by \(y = f(x)\), the x-axis, and the lines \(x = 1\) and \(x = 2\)
The area \(A\) we want to find is given by:

\[ A = \int_{1}^{2} f(x) \, dx \]

### Step 6: Use the property of integrals
We can express this integral using the property of definite integrals:

\[ \int_{1}^{2} f(x) \, dx = \int_{0}^{2} f(x) \, dx - \int_{0}^{1} f(x) \, dx \]

### Step 7: Calculate \(\int_{0}^{2} f(x) \, dx\)
To find \(\int_{0}^{2} f(x) \, dx\), we can use the fact that:

\[ \int_{0}^{2} f(x) \, dx = \int_{0}^{1} f(x) \, dx + \int_{1}^{2} f(x) \, dx \]

Let \(I = \int_{1}^{2} f(x) \, dx\). Then we have:

\[ \int_{0}^{2} f(x) \, dx = \frac{3}{7} + I \]

### Step 8: Substitute back into the area equation
Now substituting this into our area equation gives:

\[ I = \left(\frac{3}{7} + I\right) - \frac{3}{7} \]

This simplifies to:

\[ I = \frac{3}{7} + I - \frac{3}{7} \]

### Step 9: Solve for \(I\)
This implies:

\[ I = \frac{3}{7} \]

Thus, the area \(A\) bounded by \(y = f(x)\), the x-axis, and the lines \(x = 1\) and \(x = 2\) is:

\[ A = \frac{3}{7} \]

### Final Answer
The area of the region bounded by \(y = f(x)\), the x-axis, and the lines \(x = 1\) and \(x = 2\) is:

\[ \boxed{\frac{3}{7}} \]