The area bounded by the curves `y=cos^(-1)x,y =sin^(-1) x and y=-pix^(3),` where `-1lex le1`,is

Options

A. `(3pi)/(2)+1-sqrt2` sq. units
B. `(3pi)/(4)+1+sqrt2` sq. units
C. `(3pi)/(4)+2-sqrt2` sq. units
D. `(3pi)/(4)+1-sqrt2` sq. units

Explanation

To find the area bounded by the curves \( y = \cos^{-1} x \), \( y = \sin^{-1} x \), and \( y = -\pi x^3 \) for \( -1 \leq x \leq 1 \), we will follow these steps:

### Step 1: Identify the curves and their intersections
The curves we are dealing with are:
1. \( y = \cos^{-1} x \)
2. \( y = \sin^{-1} x \)
3. \( y = -\pi x^3 \)

To find the area bounded by these curves, we first need to determine the points where they intersect.

### Step 2: Find points of intersection
The points of intersection can be found by setting the equations equal to each other.

1. Set \( \cos^{-1} x = \sin^{-1} x \):
\[ \cos^{-1} x + \sin^{-1} x = \frac{\pi}{2} \]
This is true for all \( x \) in the interval \( [-1, 1] \).

2. Set \( \cos^{-1} x = -\pi x^3 \) and \( \sin^{-1} x = -\pi x^3 \) to find the other intersections.

Solving these equations can be complex, but we can check specific values:
- At \( x = 0 \):
\[ \cos^{-1}(0) = \frac{\pi}{2}, \quad \sin^{-1}(0) = 0, \quad -\pi(0)^3 = 0 \]
- At \( x = 1 \):
\[ \cos^{-1}(1) = 0, \quad \sin^{-1}(1) = \frac{\pi}{2}, \quad -\pi(1)^3 = -\pi \]
- At \( x = -1 \):
\[ \cos^{-1}(-1) = \pi, \quad \sin^{-1}(-1) = -\frac{\pi}{2}, \quad -\pi(-1)^3 = \pi \]

### Step 3: Set up the integral for the area
The area \( A \) can be calculated using the integral of the upper curve minus the lower curve. The area can be expressed as:
\[ A = \int_{a}^{b} (f(x) - g(x)) \, dx \]
where \( f(x) \) is the upper curve and \( g(x) \) is the lower curve.

### Step 4: Calculate the area
The area can be divided into segments based on the intersection points. We will integrate from \( 0 \) to \( \frac{\pi}{4} \) and from \( \frac{\pi}{4} \) to \( \frac{\pi}{2} \).

1. From \( 0 \) to \( \frac{\pi}{4} \):
\[ A_1 = \int_{0}^{\frac{\pi}{4}} (\sin^{-1} x - (-\pi x^3)) \, dx \]

2. From \( \frac{\pi}{4} \) to \( \frac{\pi}{2} \):
\[ A_2 = \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} (\cos^{-1} x - (-\pi x^3)) \, dx \]

### Step 5: Combine the areas
The total area \( A \) is given by:
\[ A = A_1 + A_2 \]

### Final Calculation
After evaluating the integrals and substituting the limits, we will arrive at the final area bounded by the curves.