The area (in sq. units) bounded by `y=max(sin^(2)x, sin^(4)x), x in [0, (pi)/(2)]`
with the x - axis, from `x=0` to `x=(pi)/(2)` is

Options

A. `pi`
B. `(pi)/(2)`
C. `(pi)/(4)`
D. `(pi)/(6)`

Answer

To find the area bounded by the curves \( y = \max(\sin^2 x, \sin^4 x) \) from \( x = 0 \) to \( x = \frac{\pi}{2} \) with the x-axis, we will follow these steps:

### Step 1: Determine the functions
We need to compare \( \sin^2 x \) and \( \sin^4 x \) over the interval \( [0, \frac{\pi}{2}] \).

### Step 2: Analyze the functions
For \( x \in [0, \frac{\pi}{2}] \):
- \( \sin^2 x \) is always greater than or equal to \( \sin^4 x \) because \( \sin^4 x = (\sin^2 x)^2 \) and \( \sin^2 x \) is between 0 and 1. Thus, squaring it will yield a smaller value.

### Step 3: Identify the maximum function
Since \( \sin^2 x \) is greater than \( \sin^4 x \) throughout the interval, we have:
\[ y = \max(\sin^2 x, \sin^4 x) = \sin^2 x \]

### Step 4: Set up the integral for the area
The area \( A \) under the curve from \( x = 0 \) to \( x = \frac{\pi}{2} \) is given by:
\[ A = \int_0^{\frac{\pi}{2}} \sin^2 x \, dx \]

### Step 5: Use the identity for integration
We can use the identity:
\[ \sin^2 x = \frac{1 - \cos(2x)}{2} \]
Thus, the integral becomes:
\[ A = \int_0^{\frac{\pi}{2}} \frac{1 - \cos(2x)}{2} \, dx \]

### Step 6: Split the integral
This can be split into two parts:
\[ A = \frac{1}{2} \int_0^{\frac{\pi}{2}} 1 \, dx - \frac{1}{2} \int_0^{\frac{\pi}{2}} \cos(2x) \, dx \]

### Step 7: Evaluate the first integral
The first integral evaluates to:
\[ \int_0^{\frac{\pi}{2}} 1 \, dx = \left[ x \right]_0^{\frac{\pi}{2}} = \frac{\pi}{2} \]

### Step 8: Evaluate the second integral
For the second integral:
\[ \int_0^{\frac{\pi}{2}} \cos(2x) \, dx = \left[ \frac{\sin(2x)}{2} \right]_0^{\frac{\pi}{2}} = \frac{\sin(\pi)}{2} - \frac{\sin(0)}{2} = 0 \]

### Step 9: Combine the results
Thus, we have:
\[ A = \frac{1}{2} \left( \frac{\pi}{2} \right) - \frac{1}{2} (0) = \frac{\pi}{4} \]

### Final Answer
The area bounded by \( y = \max(\sin^2 x, \sin^4 x) \) and the x-axis from \( x = 0 \) to \( x = \frac{\pi}{2} \) is:
\[ \boxed{\frac{\pi}{4}} \]