Statement I- Area bounded by `y=x(x-1)` and `y=x(1-x) "is" 1/3`.
Statement II- Area bounded by `y=f(x)` and `y=g(x)` "is" `|int_(a)^(b)(f(x)-g(x))dx|` is true when `f(x)` and `g(x)` lies above X-axis.(Where a and b are intersection of `y=f(x) and y=g(x))`.

Options

A. Statement I is true, Statement II is also true, Statement II is the correct explanation of Statement I.
B. Statement I is true, Statement II is also true, Statement II is not the correct explanation of Statement I.
C. Statement I is true , Statement II is false
D. Statement I is false , Statement II is true

Answer

To solve the problem, we will analyze both statements step by step.

### Step 1: Analyze Statement I
We need to find the area bounded by the curves \( y = x(x - 1) \) and \( y = x(1 - x) \).

1. **Identify the Functions:**
- The first function is \( f(x) = x(x - 1) = x^2 - x \).
- The second function is \( g(x) = x(1 - x) = x - x^2 \).

2. **Find the Points of Intersection:**
Set \( f(x) = g(x) \):
\[ x^2 - x = x - x^2 \]
Rearranging gives:
\[ 2x^2 - 2x = 0 \implies 2x(x - 1) = 0 \]
Thus, \( x = 0 \) and \( x = 1 \).

3. **Determine the Area:**
The area \( A \) between the curves from \( x = 0 \) to \( x = 1 \) is given by:
\[ A = \int_0^1 (g(x) - f(x)) \, dx \]
Substituting the functions:
\[ A = \int_0^1 \left( (x - x^2) - (x^2 - x) \right) \, dx \]
Simplifying the integrand:
\[ A = \int_0^1 (x - x^2 - x^2 + x) \, dx = \int_0^1 (2x - 2x^2) \, dx \]
Factoring out the 2:
\[ A = 2 \int_0^1 (x - x^2) \, dx \]

4. **Calculate the Integral:**
\[ \int (x - x^2) \, dx = \left( \frac{x^2}{2} - \frac{x^3}{3} \right) + C \]
Evaluating from 0 to 1:
\[ A = 2 \left[ \left( \frac{1^2}{2} - \frac{1^3}{3} \right) - \left( \frac{0^2}{2} - \frac{0^3}{3} \right) \right] \]
\[ = 2 \left( \frac{1}{2} - \frac{1}{3} \right) = 2 \left( \frac{3}{6} - \frac{2}{6} \right) = 2 \left( \frac{1}{6} \right) = \frac{1}{3} \]

### Conclusion for Statement I:
Thus, Statement I is **true** as the area bounded by the curves is indeed \( \frac{1}{3} \).

### Step 2: Analyze Statement II
Statement II claims that the area bounded by \( y = f(x) \) and \( y = g(x) \) can be expressed as:
\[ | \int_a^b (f(x) - g(x)) \, dx | \]
when \( f(x) \) and \( g(x) \) are above the x-axis, and \( a \) and \( b \) are the points of intersection.

1. **Understanding the Statement:**
The area between two curves can be calculated using the integral of the difference between the upper curve and the lower curve. The absolute value is necessary to ensure the area is positive.

2. **Graphical Representation:**
If we plot \( f(x) \) and \( g(x) \), we can see that if \( f(x) \) is above \( g(x) \) between the intersection points \( a \) and \( b \), the integral \( \int_a^b (f(x) - g(x)) \, dx \) will yield a positive value.

3. **Conclusion for Statement II:**
Therefore, Statement II is also **true**.

### Final Conclusion:
Both Statement I and Statement II are true.